Book TG · English

Index of statements

The 1136 statements of General Topology, by chapter and §. Pick a kind, filter by a word, or put the most cited first.

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I. Topological Structures

§ 1. Open sets, neighbourhoods, closed sets

Definition 1 A topological structure (or, more briefly, a topology) on a set X is a structure given by a set D of subsets of X, having the following properties (called axioms of topological str… 0AZ1
Definition 2 A topological space is a set endowed with a topological structure. 0AZ2
Definition 3 A homeomorphism of a topological space X onto a topological space X' is an isomorphism of the topological structure of X onto that of X'; that is to say, in accordance with the gen… 0AZ3
Example If X and X' are two discrete spaces, any bijection of X onto X' is a homeomorphism. 0AZ4
Definition 4 Let X be a topological space and A any subset of X. A neighbourhood of A is any subset of X which contains an open set containing A. The neighbourhoods of a subset x consisting of… ↩ 1 0AZ5
Proposition 1 A set is a neighbourhood of each of its points if and only if it is open. ↩ 6 0AZ6
Proposition 2 If to each element x of a set X there corresponds a set 𝔅(x) of subsets of X such that the properties (VI), (V{II}), (V{III}) and (V{IV}) are satisfied, then there is a unique topo… ↩ 1 0AZ7
Example We may define a topology on the set Q of rational numbers by taking for open sets all unions of bounded open intervals; the set of these subsets certainly satisfies (OI), and to se… 0AZ8
Definition 5 In a topological space X, a fundamental system of neighbourhoods of a point x (resp. of a subset A of X) is any set 𝔊 of neighbourhoods of x (resp. A) such that for each neighbourh… ↩ 1 0AZ9
Definition 6 A base of the topology of a topological space X is any set 𝔅 of open subsets of X such that every open subset of X is the union of sets belonging to 𝔅. 0AZA
Proposition 3 If X is a topological space, then for a set 𝔅 of open subsets of X to be a base of the topology of X it is necessary and sufficient that for each x ∈ X the set of V ∈ 𝔅 such that x… ↩ 2 0AZB
Example 1 The discrete topology has as a base the set of subsets of X which consist of a single point. 0AZC
Example 2 The set of bounded open intervals is by definition a base of the topology of the rational line (no. 2). \ Likewise, the set of bounded open intervals is a base of the topology of t… 0AZD
Definition 7 In a topological space X, the complements of the open sets of X are called closed sets. 0AZE
Definition 8 A family (Aᵢ)_i ∈ I of subsets of a topological space X is said to be locally finite if for each x ∈ X there is a neighbourhood V of x such that V ∩ Aᵢ = ∅ for all but a finite num… 0AZF
Proposition 4 The union of a locally finite family of closed subsets of a topological space X is closed in X. ↩ 6 0AZG
Definition 9 In a topological space X, a point x is said to be an interior point of a subset A of X if A is a neighbourhood of x. The set of interior points of A is called the interior of A and… 0AZH
Remark The interior of a non-empty set can be empty; this is the case for a set consisting of a single point which is not open, for example on the rational line \ (or the real line) . 0AZI
Definition 10 The closure of a subset A of a topological space X is the set of all points x ∈ X such that every neighbourhood of x meets A, and is denoted by overlineA. ↩ 1 0AZJ
Proposition 5 Let A be an open set in X; then for every subset B of X we have ↩ 3 0AZK
Definition 11 In a topological space X, a point x is said to be a frontier point of a set A if x lies in the closure of A and in the closure of overlineCA. The set of frontier points of A is cal… 0AZL
Definition 12 A subset A of a topological space X is said to be dense in X (or simply dense, if there is no ambiguity about X) if overlineA = X, i.e. if every non-empty open set U of X meets A. 0AZM
Example \ We shall see in Chapter IV, § 1 that the set of rational numbers and its complement are dense on the real line. \ In a discrete space X the only dense subset of X is X itself. On… 0AZN
Proposition 6 If 𝔅 is a base of the topology of a topological space X, there is a dense set D in X such that Card(D) ≤ Card(𝔅). ↩ 1 0AZO

§ 2. Continuous functions

Definition 1 A mapping f of a topological space X into a topological space X' is said to be continuous at a point x₀ ∈ X if, given any neighbourhood V' of f(x₀) in X', there is a neighbourhood… ↩ 2 0AZY
Proposition 1 Let f be a mapping of a topological space X into a topological space X'. If f is continuous at x, and if x lies in the closure of a subset A of X, then f(x) lies in the closure of… ↩ 1 0AZZ
Proposition 2 Let X, X', X'' be three topological spaces; let f be a mapping of X into X', continuous at x ∈ X; let g be a mapping of X' into X'', continuous at f(x). Then the composition h = g… ↩ 3 0B00
Definition 2 A mapping of a topological space X into a topological space X' is said to be continuous on X (or just continuous) if it is continuous at each point of X. 0B01
Example 1 The identity mapping of a topological space X onto itself is continuous. 0B02
Example 2 A constant map of a topological space into a topological space is continuous. 0B03
Example 3 Every mapping of a discrete space into a topological space is continuous. 0B04
Theorem 1 Let f be a mapping of a topological space X into a topological space X'. Then the following statements are equivalent: ↩ 25 0B05
Remarks 1) Let 𝔅 be a base (§ 1, no. 3) of the topology of X'; then for f : X → X' to be continuous, it is necessary and sufficient that overlinef⁻¹(U') is open in X for every U' ∈ 𝔅. 0E95
Example Let a be any rational number. The mapping x → a + x of the rational line Q into itself is continuous on Q, for the inverse image under this mapping of an open interval ]b, c[ is th… 0B06
Example \ The mapping f : x → 1/(1 + x²) of R into itself is continuous, but f(R) is the half-open interval ]0, 1], which is neither open nor closed in R. \ 0B07
Theorem 2 1) If f : X → X' and g : X' → X'' are two continuous mappings, then g ∘ f : X → X'' is continuous. ↩ 4 0B08
Remark 1 It is possible to have a continuous bijection of a topological space X onto a topological space X' which is not bicontinuous: for example, take X' to be the rational line Q, and X… ↩ 1 0B09
Remark 2 To verify that a continuous bijection f : X → X' is a homeomorphism, it is enough to show that for each x ∈ X and each neighbourhood V of x, f(V) is a neighbourhood of f(x) in X'. 0B0A
Remark 3 Let X be a topological space, and for each x ∈ X let 𝔅(x) be the set of all neighbourhoods of x. Let x₀ be a point of X; for each x ∈ X, define a set 𝔅₀(x) of subsets of X as follo… 0B0B
Definition 3 Given two topologies 𝔗₁, 𝔗₂ on the same set X, we say that 𝔗₁ is finer than 𝔗₂ (and that 𝔗₂ is coarser than 𝔗₁) if, denoting by Xᵢ the set X with the topology 𝔗ᵢ (i = 1, 2), the id… 0B0C
Proposition 3 Given two topologies 𝔗₁, 𝔗₂ on a set X, the following statements are equivalent: a) 𝔗₁ is finer than 𝔗₂. b) For each x ∈ X, each neighbourhood of x for 𝔗₂ is a neighbourhood of x f… ↩ 1 0B0D
Example \ In Hilbert space H consisting of sequences x = (xₙ) of real numbers such that |x|² = ∑ₙ₌₀^∞ xₙ² < +∞, the neighbourhoods of a point x₀ in the strong topology on H are the sets wh… 0B0E
Remark 1 In the ordered set of all topologies on a set X, the topology in which the only open sets are ∅ and X is the coarsest and the discrete topology is the finest. 0B0F
Remark 2 The finer the topology, the more open sets, closed sets and neighbourhoods; the finer the topology, the smaller (resp. the larger) the closure (resp. the interior) of a set; the fi… 0B0G
Remark 3 If f : X → X' is a continuous mapping, it remains continuous if the topology of X is replaced by a finer topology and the topology of X' is replaced by a coarser topology (no. 1, T… 0B0H
Proposition 4 Let X be a set, let (Yᵢ)_i ∈ I be a family of topological spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let S be the set of subsets of X of the form overlinefᵢ⁻¹(Uᵢ)… ↩ 11 0B0I
Proposition 5 Let X be a set, (Zᵢ)_i ∈ I a family of topological spaces, (J_λ)_λ ∈ L a partition of I and (Y_λ)_λ ∈ L a family of sets indexed by L. Also for each λ ∈ L let h_λ be a mapping of X… ↩ 2 0B0J
Example 1 Inverse image of a topology. Let X be a set, Y a topological space, f a mapping of X into Y; the coarsest topology 𝔗 on X for which f is continuous is called the inverse image unde… ↩ 2 0B0K
Example 2 Least upper bound of a set of topologies. Every family (𝔗ᵢ)_i ∈ I of topologies on a set X has a least upper bound 𝔗 in the ordered set of all topologies on X, i.e. there exists a… ↩ 1 0B0L
Example 3 Product topology. Let (Xᵢ)_i ∈ I be a family of topological spaces. The coarsest topology on the product set X = ∏_i ∈ I Xᵢ for which the projections prᵢ : X → Xᵢ are continuous ma… ↩ 2 0B0M
Proposition 6 Let X be a set, let (Yᵢ)_i ∈ I be a family of topological spaces, and for each i ∈ I let fᵢ be a mapping of Yᵢ into X. Let D be the set of subsets U of X such that overlinefᵢ⁻¹(U)… ↩ 6 0B0N
Corollary Under the hypotheses of Proposition 6, a subset F of X is closed in the topology 𝔗 if and only if overlinefᵢ⁻¹(F) is closed in Yᵢ for each i ∈ I. 0B0O
Proposition 7 Let X be a set, (Zᵢ)_i ∈ I a family of topological spaces, (J_λ)_λ ∈ L a partition of I, and (Y_λ)_λ ∈ L a family of sets indexed by L. Also, for each λ ∈ L let h_λ be a mapping of… ↩ 1 0B0P
Proposition 8 Let X be a set, (X_λ)_λ ∈ L a family of subsets of X. Suppose each X_λ carries a topology 𝔗_λ such that, for each pair of indices (λ, μ): 1) X_λ ∩ X_μ is open (resp. closed) in eac… ↩ 1 0B0Q
Proposition 9 Suppose that the h_μ λ are homeomorphisms and that each A_λ μ is open (resp. closed) in X_λ; then each φ(X_λ) is open (resp. closed) in X/R and the restriction of φ to X_λ is a hom… 0B0R

§ 3. Subspaces, quotient spaces

Definition 1 Let A be a subset of a topological space X. The topology induced on A by the topology of X is that in which the open sets are the intersections with A of open sets of X. The set A… ↩ 1 0B13
Example The topology induced on the set Z of rational integers by the topology of the rational line is the discrete topology, for the intersection of Z and the open interval ]n - 1/2, n +… 0B14
Proposition 1 If A and B are two subsets of a topological space X, and B ⊂ A, then the closure of B in the subspace A is the intersection with A of the closure overlineB of B in X. ↩ 1 0B15
Corollary A subset B of A is dense in A if and only if overlineB = overlineA in X (i.e. if and only if A ⊂ overlineB). 0B16
Proposition 2 Let A be a dense subset of a topological space X; then for each x ∈ A and each neighbourhood V of x relative to A, the closure overlineV of V in X is a neighbourhood of x in X. ↩ 1 0B17
Proposition 3 Let (Aᵢ)_i ∈ I be a family of subsets of a topological space X, such that one of the following properties holds: a) The interiors of the Aᵢ cover X. b) (Aᵢ)_i ∈ I is a locally fini… ↩ 6 0B18
Remark Let (Uᵢ)_i ∈ I be an open covering of a topological space X, and for each i ∈ I let 𝔅ᵢ be a base of the topology of the subspace Uᵢ of X; then it is clear that 𝔅 = ⋃_i ∈ I 𝔅ᵢ is a… 0B19
Proposition 4 Let (Aᵢ)_i ∈ I be a family of subsets of a topological space X whose interiors cover X, or which is a locally finite closed covering of X. Let f be a mapping of X into a topologica… 0B1A
Definition 2 A subset L of a topological space X is said to be locally closed at a point x ∈ L if there is a neighbourhood V of x in X such that L ∩ V is a closed subset of the subspace V. L is… ↩ 1 0B1B
Remark Let F be a subset of X such that for each point x of X there is a neighbourhood V of x such that V ∩ F is closed in the subspace V; then it follows from Proposition 3 of no. 1 that… 0B1C
Proposition 5 For a subset L of a topological space X, the following properties are equivalent: ↩ 2 0B1D
Corollary Let f : X → X' be a continuous map and L' a locally closed subset of X'; then f⁻¹(L') is locally closed in X. 0B1E
Definition 3 Let X be a topological space, R an equivalence relation on X. The quotient space of X by R is the quotient set X/R with the topology which is the quotient of the topology of X by t… 0B1F
Proposition 6 Let X be a topological space, R an equivalence relation on X, φ the canonical mapping of X onto X/R; then a mapping f of X/R into a topological space Y is continuous if and only if… ↩ 6 0B1G
Example Consider the equivalence relation x ≡ y pmod1 on the real line R; the quotient space of R by this relation is called the one-dimensional torus and is denoted by T. The equivalence… 0B1H
Corollary Let X, Y be two topological spaces, R (resp. S) an equivalence relation on X (resp. Y), and let f : X → Y be a continuous mapping which is compatible with the equivalence relations… ↩ 1 0B1I
Proposition 7 (Transitivity of quotient spaces) Let R and S be two equivalence relations on a topological space X such that R implies S, and let S/R be the quotient equivalence relation on the quotient space X/R (Set Theory R, §… ↩ 2 0B1J
Proposition 8 Let f = ψ ∘ g ∘ φ be the canonical decomposition of a continuous mapping f : X → Y, and let R denote the equivalence relation f(x) = f(y). Then the following three conditions are e… ↩ 1 0B1K
Example Let X be a topological space, (Xᵢ)_i ∈ I a covering of X, Y the sum of the subspaces Xᵢ of X; then there is a partition (Yᵢ)_i ∈ I of Y into subspaces which are both open and close… 0B1L
Proposition 9 Let f : X → Y be a continuous surjection, and let R denote the equivalence relation f(x) = f(y). If there is a continuous section s : Y → X associated with f (Set Theory, Chapter I… 0B1M
Proposition 10 The canonical bijection h : A/R_A → f(A) is continuous. Furthermore, the following three statements are equivalent: ↩ 5 0B1N
Corollary 1 If A is an open (resp. closed) subset of X which is saturated with respect to R, then the canonical mapping h : A/R_A → f(A) is a homeomorphism. 0B1O
Corollary 2 If there is a continuous mapping u : X → A such that u(x) is congruent to x mod R for each x ∈ X, then f(A) = X/R and the canonical mapping h : A/R_A → X/R is a homeomorphism. ↩ 1 0B1P
Example Let R denote the equivalence relation x ≡ y pmod1 on the real line R (no. 4, Example) and let A denote the closed interval [0, 1]; A contains at least one point of each equivalence… 0B1Q

§ 4. Product of topological spaces

Definition 1 Given a family (Xᵢ)_i ∈ I of topological spaces, the product space of this family is the product set X = ∏_i ∈ I Xᵢ with the topology which is the product of the topologies of the… 0B27
Example \ The product Rⁿ of n spaces identical with the real line R is called real number space of n dimensions; R² is also called the real plane (cf. Chapter VI, § 1). Likewise, starting… 0B28
Proposition 1 Let f = (fᵢ) be a mapping of a topological space Y into a product space X = ∏_i ∈ I Xᵢ. Then f is continuous at a point a ∈ Y if and only if fᵢ is continuous at a for each i ∈ I. ↩ 11 0B29
Corollary 1 Let (Xᵢ)_i ∈ I, (Yᵢ)_i ∈ I be two families of topological spaces with the same set of indices. For each i ∈ I, let fᵢ be a mapping of Xᵢ into Yᵢ. In order that the product mapping… ↩ 5 0B2A
Corollary 2 Let X, Y be two topological spaces. In order that a mapping f : X → Y should be continuous it is necessary and sufficient that the mapping g : x → (x, f(x)) is a homeomorphism of X… ↩ 1 0B2B
Proposition 2 (Associativity of topological products) Let (Xᵢ)_i ∈ I be a family of topological spaces, (Jₓ)_x ∈ K a partition of the set I, and for each x ∈ K let X'ₓ = ∏_i ∈ Jₓ Xᵢ be the product of the spaces Xᵢ for i ∈ J_K. Then th… ↩ 4 0B2C
Corollary Let σ be a permutation of the set I. Then the mapping (xᵢ) → (x_σ(i)) is a homeomorphism of ∏_i ∈ I Xᵢ onto ∏_i ∈ I X_σ(i). 0B2D
Proposition 3 Let X be a set, (Yᵢ)_i ∈ I a family of topological spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let f be the mapping x → (fᵢ(x)) of X into Y = ∏_i ∈ I Yᵢ, and let 𝔗… ↩ 4 0B2E
Corollary For each i ∈ I let Aᵢ be a subspace of Yᵢ. Then the topology induced on A = ∏_i ∈ I Aᵢ by the product topology on ∏_i ∈ I Yᵢ is the product of the topologies of the subspaces Aᵢ. 0B2F
Proposition 4 Let X₁, X₂ be two topological spaces; then for each a₁ ∈ X₁, the mapping x₂ → (a₁, x₂) is a homeomorphism of X₂ onto the subspace a₁ × X₂ of X₁ × X₂. ↩ 2 0B2G
Corollary The section A(x₁) of an open (resp. closed) set A of the product X₁ × X₂ at an arbitrary point x₁ ∈ X₁ (*) is open (resp. closed) in X₂. ↩ 2 0B2H
Proposition 5 The projection of an open set U of the product X₁ × X₂ onto either factor is an open set. ↩ 4 0B2I
Remark 1 The projection of a closed subset of X₁ × X₂ onto a factor need not be closed. For example, in the rational plane Q², the hyperbola whose equation is x₁ x₂ = 1 is a closed set, but… ↩ 1 0B2J
Proposition 6 Let X₁, X₂, Y be three topological spaces, f a mapping of the product space X₁ × X₂ into Y. If f is continuous at the point (a₁, a₂) then the partial mapping x₂ → f(a₁, x₂) of X₂ i… 0B2K
Remark 2 It is possible for all the partial mappings determined by a map f : X₁ × X₂ → Y to be continuous without f being continuous on X₁ × X₂ (cf. Chapter IX, § 5, Exercise 23). \ For exa… ↩ 1 0B2L
Proposition 7 In a product space ∏_i ∈ I Xᵢ, the closure of a product of sets ∏_i ∈ I Aᵢ is the same as the product ∏_i ∈ I overlineAᵢ of their closures. ↩ 3 0B2M
Corollary A product ∏_i ∈ I Aᵢ of non-empty sets is closed in the product space ∏_i ∈ I Xᵢ if and only if Aᵢ is closed in Xᵢ for each i ∈ I. ↩ 2 0B2N
Proposition 8 Let a = (aᵢ) be any point of a product space X = ∏_i ∈ I Xᵢ; then the set D of points x ∈ X such that prᵢ x = aᵢ except for a finite number of indices i is dense in X. ↩ 2 0B2O
Proposition 9 Let I be a directed set and J a cofinal subset of I. Let (X_α, f_αβ) be an inverse system of topological spaces indexed by I; let X = varprojlim X_α and let f_α : X → X_α be the ca… ↩ 4 0B2P
Corollary Let A be a subset of X and let A_α denote f_α(A) for each α ∈ I. Then: ↩ 2 0B2Q
Example Let I be a directed set and (X_α)_α ∈ I a family of subsets of a set Y, such that X_α ⊃ X_β whenever α ≤ β. For each α ∈ I let 𝔗_α be a topology on X_α such that 𝔗_β is finer than… 0B2R

§ 5. Open mappings and closed mappings

Definition 1 Let X, X' be two topological spaces. A mapping f : X → X' is open (resp. closed) if the image under f of each open (resp. closed) set of X is open (resp. closed) in X'. ↩ 1 0B32
Example 1 Let A be a subspace of a topological space X. Then the canonical injection j : A → X is open (resp. closed) if and only if A is open (resp. closed) in X (§ 3, no. 1). 0B33
Example 2 For a bijection f of a topological space X onto a topological space X' to be a homeomorphism it is necessary and sufficient that f is continuous and open, or continuous and closed. 0B34
Example 3 Let f be a surjection of a set X onto a topological space X'; if we give X the topology which is the inverse image under f of the topology of X' (§ 2, no. 3, Example 1), then f is… ↩ 1 0B35
Example 4 In a product space X = ∏_i ∈ I Xᵢ, each projection prᵢ : X → Xᵢ is a continuous open mapping, but is not necessarily closed (§ 4, no. 2, Proposition 5). ↩ 1 0B36
Example 5 \ A holomorphic function on an open subset A of C is an open mapping of A into C. \ 0B37
Example 6 Let X, X' be two topological spaces and f a continuous, but not bicontinuous, bijection of X onto X'. Then the inverse bijection g : X' → X is an open and closed mapping of X' onto… 0B38
Proposition 1 Let X, X', X'' be three topological spaces, and let f : X → X', \ g : X' → X'' be two mappings. Then: ↩ 4 0B39
Proposition 2 Let X, Y be two topological spaces, f a mapping of X into Y. For each subset T of Y let f_T denote the mapping of f⁻¹(T) into T which agrees with f on f⁻¹(T). ↩ 3 0B3A
Corollary Let (T(i))_i ∈ I be a family of subsets of Y whose interiors cover Y, or which is a locally finite closed covering of Y. If f : X → Y is continuous and if each of the f_T(i) is a h… 0B3B
Definition 2 An equivalence relation R on a topological space X is said to be open (resp. closed) if the canonical mapping of X onto X/R is open (resp. closed). 0B3C
Example 1 Let X be a topological space, Γ a group of homeomorphisms of X onto itself, and let R be the equivalence relation 0B3D
Example 2 Let X be the sum of a family (Xᵢ) of subspaces of X, and let X/R be the space obtained by pasting together the Xᵢ along open subsets Aᵢₓ by means of bijections hᵢₓ (§ 2, no. 5); an… ↩ 1 0B3E
Example 3 With the notation of Example 2, suppose now that the Aᵢₓ are closed and the h_x! are homeomorphisms; further suppose that for each index i there is only a finite number of indices… ↩ 1 0B3F
Proposition 3 Let X, Y be two topological spaces, let f : X → Y be a continuous mapping, let R be the equivalence relation f(x) = f(y) on X, and let X xrightarrowp X/R xrightarrowh f(X) xrightar… ↩ 4 0B3G
Proposition 4 Let R be an open (resp. closed) equivalence relation on a topological space X, and f the canonical mapping X → X/R. Let A be a subset of X and suppose that one of the following two… ↩ 3 0B3H
Proposition 5 Let X, Y be two topological spaces, f a mapping of X into Y, 𝔅 a base of the topology of X. Then the following statements are equivalent: ↩ 6 0B3I
Proposition 6 Let R be an equivalence relation on a topological space X; then the following three conditions are equivalent: ↩ 1 0B3J
Proposition 7 Let R be an open equivalence relation on a topological space X, and let φ : X → X/R be the canonical mapping. If A is any subset of X which is saturated with respect to R, then the… ↩ 1 0B3K
Proposition 8 Let (Xᵢ)_i ∈ I, (Yᵢ)_i ∈ I be two families of topological spaces indexed by the same set I. For each i ∈ I let fᵢ be an open mapping of Xᵢ into Yᵢ, and suppose that fᵢ is surjectiv… ↩ 1 0B3L
Corollary Let (Xᵢ)_i ∈ I be a family of topological spaces, and for each i ∈ I let Rᵢ be an equivalence relation on Xᵢ, and let fᵢ be the canonical mapping Xᵢ → Xᵢ/Rᵢ. Let R be the equivalen… ↩ 4 0B3M
Proposition 9 Let X, X' be two topological spaces. A necessary and sufficient condition for a mapping f : X → X' to be continuous and closed is that f(Ā) = f(A) for every subset A of X. ↩ 1 0B3N
Proposition 10 Let R be an equivalence relation on a topological space X. Then R is closed if and only if every equivalence class M mod R possesses a fundamental system of neighbourhoods which ar… ↩ 5 0B3O
Remark Proposition 10 implies the following: if R is closed and if φ denotes the canonical mapping X → X/R, then for each x ∈ X and each neighbourhood U of the equivalence class of x in X… 0B3P

§ 6. Filters

Definition 1 A filter on a set X is a set 𝔉 of subsets of X which has the following properties: 0B3X
Definition 2 Given two filters 𝔉, 𝔉' on the same set X, 𝔉' is said to be finer than 𝔉, or 𝔉 is coarser than 𝔉', if 𝔉 ⊂ 𝔉'. If also 𝔉 ≠ 𝔉', then 𝔉' is said to be strictly finer than 𝔉, or 𝔉 stri… 0B3Y
Proposition 1 A necessary and sufficient condition that there should exist a filter on X containing a set 𝔊 of subsets of X is that no finite subset of 𝔊 has an empty intersection. ↩ 7 0B3Z
Example Let S be any set of subsets of a set X, and let 𝔗 be the topology on X generated by S (S 2, no. 3, Example II). Since the set of finite intersections of sets of S is a base of 𝔗, i… 0B40
Corollary 1 Let 𝔉 be a filter on a set X, and A a subset of X. Then there is a filter 𝔉' which is finer than 𝔉 and such that A ∈ 𝔉', if and only if A meets all the sets of 𝔉. ↩ 3 0B41
Corollary 2 A set Φ of filters on a non-empty set X has a least upper bound in the set of all filters on X if and only if, for all finite sequences (𝔉ᵢ)_1 ≤ i ≤ n of elements of Φ and all Aᵢ ∈… ↩ 3 0B42
Corollary 3 The ordered set of all filters on a non-empty set X is inductive. ↩ 1 0B43
Proposition 2 Let 𝔅 be a set of subsets of a set X. Then the set of subsets of X which contain a set of 𝔅 is a filter if and only if 𝔅 has the following two properties: (BI) The intersection of… 0B44
Definition 3 A set 𝔅 of subsets of a set X which satisfies axioms (BI) and (B{II}) is said to be a base of the filter it generates. Two filter bases are said to be equivalent if they generate t… ↩ 1 0B45
Proposition 3 A subset 𝔅 of a filter 𝔉 on X is a base of 𝔉 if and only if every set of 𝔉 contains a set of 𝔅. ↩ 5 0B46
Proposition 4 On a set X, a filter 𝔉' with base 𝔅' is finer than a filter 𝔉 with base 𝔅 if and only if every set of 𝔅 contains a set of 𝔅'. ↩ 2 0B47
Corollary Two filter bases 𝔅, 𝔅' on a set X are equivalent if and only if every set of 𝔅 contains a set of 𝔅' and every set of 𝔅' contains a set of 𝔅. ↩ 1 0B48
Definition 4 An ultrafilter on a set X is a filter 𝔉 such that there is no filter on X which is strictly finer than 𝔉 (in other words, a maximal element in the ordered set of all filters on X). 0B49
Theorem 1 If 𝔉 is any filter on a set X, there is an ultrafilter finer than 𝔉. ↩ 3 0B4A
Proposition 5 Let 𝔉 be an ultrafilter on a set X. If A and B are two subsets of X such that A ∪ B ∈ 𝔉, then either A ∈ 𝔉 or B ∈ 𝔉. ↩ 5 0B4B
Corollary If the union of a finite sequence (Aᵢ)_1 ≤ i ≤ n of subsets of X belongs to an ultrafilter 𝔉, then at least one of the Aᵢ belongs to 𝔉. ↩ 2 0B4C
Proposition 6 Let 𝔊 be a subbase of a filter on a set X. If for each subset Y of X we have either Y ∈ 𝔊 or complement Y ∈ 𝔊, then 𝔊 is an ultrafilter on X. ↩ 1 0B4D
Remark If X contains at least two elements, there are at least two distinct ultrafilters on X, and therefore the ordered set of filters on X has no greatest element. 0B4E
Proposition 7 Every filter 𝔉 on a set X is the intersection of the ultrafilters finer than 𝔉. ↩ 2 0B4F
Proposition 8 Let 𝔉 be a filter on a set X and A a subset of X. Then the trace 𝔉_A of 𝔉 on A is a filter if and only if each set of 𝔉 meets A. ↩ 1 0B4G
Definition 5 Let A be a subset of a set X and 𝔉 a filter on X. If the trace of 𝔉 on A is a filter on A, this filter is said to be induced by 𝔉 on A. 0B4H
Example Let X be a topological space, A a subset of X, x a point of X. In order that the trace on A of the neighbourhood filter 𝔅 of x should be a filter on A, it is necessary and sufficie… 0B4I
Proposition 9 An ultrafilter U on a set X induces a filter on a subset A of X if and only if A ∈ U; and if this condition is satisfied then U_A is an ultra-filter on A. 0B4J
Proposition 10 If 𝔅 is an ultrafilter base on a set X and if f is a mapping of X into a set X', than f(𝔅) is an ultrafilter base on X'. ↩ 2 0B4K
Definition 6 Given a filter 𝔉ᵢ on each set Xᵢ of a family of sets (Xᵢ)_i ∈ I, the product of the filters 𝔉ᵢ is the filter on X = ∏_i ∈ I Xᵢ which has as a base the set of subsets of X of the fo… 0B4L
Definition 7 Let (xₙ)_n ∈ N be an infinite sequence of elements of a set X. The elementary filter associated with the sequence (xₙ) is the filter generated by the image of the Fréchet filter (n… 0B4M
Proposition 11 If a filter 𝔉 has a countable base, it is the intersection of the elementary filters which are finer than 𝔉. ↩ 2 0B4N
Remark A filter which is coarser than a filter with a countable base need not possess a countable base; for example, if X is an uncountable infinite set, then the filter consisting of the… 0B4O
Remark 1 Every mapping f of a subset M of X into X', where M belongs to 𝔉, is equivalent mod S to a mapping f₁ of X into X' (which justifies the above terminology): it is sufficient to exte… 0B4P
Remark 2 The characteristic functions φ_M and φ_N of two subsets M and N of X have the same germ with respect to 𝔉 if and only if M and N have the same germ with respect to 𝔉. 0B4Q

§ 7. Limits

Definition 1 Let X be a topological space and 𝔉 a filter on X. A point x ∈ X is said to be a limit point (or simply a limit) of 𝔉, if 𝔉 is finer than the neighbourhood filter 𝔅(x) of x; 𝔉 is al… ↩ 1 0B5B
Proposition 1 A filter base 𝔅 on a topological space X converges to x if and only if every set of a fundamental system of neighbourhoods of x contains a set of 𝔅. 0B5C
Proposition 2 A filter 𝔉 on a topological space X converges to a point x if and only if every ultrafilter which is finer than 𝔉 converges to x. 0B5D
Definition 2 In a topological space X, a point x is a cluster point of a filter base 𝔅 on X if it lies in the closure of all the sets of 𝔅. 0B5E
Proposition 3 A point x is a cluster point of a filter base B if and only if every set of a fundamental system of neighbourhoods of x meets every set of B. 0B5F
Proposition 4 A point x is a cluster point of a filter 𝔉 if and only if there is a filter finer than 𝔉 which converges to x. ↩ 1 0B5G
Corollary An ultrafilter 𝔘 converges to a point x if and only if x is a cluster point of 𝔘. ↩ 1 0B5H
Proposition 5 The set of cluster points of a filter base on a topological space X is closed in X. ↩ 1 0B5I
Proposition 6 Let 𝔅 be a filter base on a subset A of a topological space X. Then every cluster point of 𝔅 in X belongs to overlineA; and conversely every point of overlineA is a limit point of… 0B5J
Remark A filter on a topological space need have no cluster points (and a fortiori no limit points); for example, in an infinite discrete space the filter of complements of finite subsets… 0B5K
Definition 3 Let f be a mapping of a set X into a topological space Y, and let 𝔉 be a filter on X. A point y ∈ Y is said to be a limit point (or simply a limit) (resp. cluster point) of f with… 0B5L
Proposition 7 A point y ∈ Y is a limit of f with respect to the filter 𝔉 if and only if, for each neighbourhood V of y in Y, there is a set M ∈ 𝔉 such that f(M) ⊂ V (i.e. overlinef(V) ∈ 𝔉 for ea… ↩ 2 0B5M
Example 1 A sequence of points (xₙ)_n ∈ N of a topological space is a mapping n → xₙ of N into X. In analysis one frequently uses the notions of limit point and cluster point of such a mappi… 0B5N
Example 2 More generally, let f be a mapping of a directed set A into a topological space X. If x ∈ X is a limit (resp. cluster point) of f with respect to the section filter of A then x is… 0B5O
Proposition 8 Let f be a mapping of a set X into a topological space Y; then y ∈ Y is a cluster point of f with respect to 𝔉 if and only if there is a filter 𝔊 on X which is finer than 𝔉 and suc… 0B5P
Remark If y ∈ Y is a limit (resp. cluster point) of a mapping f : X → Y with respect to a filter 𝔉 on X, then y is also a limit (resp. cluster point) of every function g : X → Y which has… 0B5Q
Proposition 9 A mapping f of a topological space X into a topological space Y is continuous at a point a ∈ X if and only if lim_i ∈ I f(x) = f(a). ↩ 6 0B5R
Corollary 1 Let X, Y be two topological spaces, f a mapping of X into Y which is continuous at a point a ∈ X; then, for every filter base 𝔅 on X which converges to a, the filter base f(𝔅) conv… ↩ 6 0B5S
Corollary 2 Let g be a mapping of a set Z into a topological space X, which has a limit a with respect to a filter 𝔉 on Z; then if the map f : X → Y is continuous at a, the composition f ∘ g h… 0B5T
Proposition 10 Let X be a set, let (Yᵢ)_i ∈ I be a family of topological spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let X be given the coarsest topology 𝔗 for which the fᵢ are c… ↩ 3 0B5U
Corollary 1 A filter 𝔉 on a product space X = ∏_i ∈ I Xᵢ converges to a point x if and only if for each i ∈ I the filter base prᵢ(𝔉) converges to prᵢ(x). ↩ 5 0B5V
Corollary 2 Let f = (fᵢ) be a mapping of a set X into a product space Y = ∏_i ∈ I Yᵢ. Then f has a limit y = (yᵢ) with respect to a filter 𝔉 on X if and only if for each i ∈ I fᵢ has limit yᵢ… 0B5W
Proposition 11 Let R be an open equivalence relation on a topological space X and let φ be the canonical mapping X → X/R. Then for each x ∈ X and each filter base 𝔅' on X/R which converges to φ(x… ↩ 1 0B5X

§ 8. Hausdorff spaces and regular spaces

Proposition 1 Let X be a topological space. Then the following statements are equivalent: ↩ 13 0B65
Definition 1 A topological space satisfying the conditions of Proposition 1 is called a Hausdorff or separated space; the topology of such a space is said to be a Hausdorff topology. 0B66
Example Any discrete space is Hausdorff. The rational line Q is Hausdorff, for if x, y are two rational numbers such that x < y then there is a rational number z such that x < z < y, and t… 0B67
Proposition 2 Let f, g be two continuous mappings of a topological space X into a Hausdorff space Y; then the set of all x ∈ X such that f(x) = g(x) is closed in X. ↩ 16 0B68
Corollary 1 (Principle of extension of identities) Let f, g be two continuous mappings of a topological space X into a Hausdorff space Y. If f(x) = g(x) at all points of a dense subset of X, then f = g. ↩ 10 0B69
Corollary 2 If f is a continuous mapping of a topological space X into a Hausdorff space Y, then the graph of f is closed in X × Y. ↩ 4 0B6A
Proposition 3 Let (xᵢ)_1 ≤ i ≤ n be a finite family of distinct points of a Hausdorff space X; then each xᵢ has a neighbourhood Vᵢ in X such that the Vᵢ (1 ≤ i ≤ n) are mutually disjoint. 0B6B
Corollary Every finite Hausdorff space is discrete. 0B6C
Proposition 4 Every finite subset of a Hausdorff space is closed. ↩ 1 0B6D
Proposition 5 Let X be a topological space and suppose that for each pair of distinct points x, y of X there is a continuous mapping f of X into a Hausdorff space X' such that f(x) ≠ f(y). Then… ↩ 5 0B6E
Corollary Every topology which is finer than a Hausdorff topology is Hausdorff. 0B6F
Proposition 6 If every point of a topological space X has a closed neighbourhood which is a Hausdorff subspace of X, then X is Hausdorff. ↩ 2 0B6G
Proposition 7 Every product of Hausdorff spaces is Hausdorff. Conversely, if a product of non-empty spaces is Hausdorff, then each factor is a Hausdorff space. ↩ 5 0B6H
Corollary 1 Let X be a set, let (Yᵢ)_i ∈ I be a family of Hausdorff topological spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let X carry the coarsest topology 𝔗 for which the f… 0B6I
Corollary 2 Let (X_α, f_α β) be an inverse system of topological spaces. If the X_α are Hausdorff, then X = varprojlim X_α is Hausdorff and is a closed subspace of ∏_α X_α. ↩ 3 0B6J
Proposition 8 A necessary condition for the quotient space X/R to be Hausdorff is that the graph C of R is closed in X × X. If the equivalence relation R is open, this condition is also sufficie… ↩ 7 0B6K
Proposition 9 If f is a continuous mapping of a topological space X into a Hausdorff space X', and R is the equivalence relation f(x) = f(y), then the quotient space X/R is Hausdorff. 0B6L
Proposition 10 If X is a Hausdorff space, and if X has a continuous section s with respect to the equivalence relation R, then X/R is Hausdorff and s(X/R) is closed in X. ↩ 1 0B6M
Proposition 11 The following properties of a topological space X are equivalent: 0B6N
Definition 2 A topological space is said to be regular if it is Hausdorff and satisfies axiom (O{III}); its topology is then said to be regular. 0B6O
Proposition 12 Every subspace of a regular space is regular. 0B6P
Proposition 13 If every point x of a topological space X has a closed neighbourhood which is a regular subspace of X, then X is regular. ↩ 2 0B6Q
Remark 1 There are examples of non-Hausdorff spaces in which every point has a regular neighbourhood (Exercise 7). 0B6R
Remark 2 There are spaces which are Hausdorff but not regular (Exercise 20). 0B6S
Remark 3 A topology which is finer than a regular topology need not be regular (Exercise 20). 0B6T
Theorem 1 Let X be a topological space, A a dense subset of X, f : A → Y a mapping of A into a regular space Y. A necessary and sufficient condition for f to extend to a continuous mapping o… ↩ 3 0B6U
Corollary Let 𝔉₁ be a filter on a set X₁, and 𝔉₂ a filter on a set X₂; let 𝔉₁ × 𝔉₂ be the product filter (§ 6, no. 7) on X = X₁ × X₂, and let f be a mapping of X into a regular space Y. Supp… 0B6V
Proposition 14 Let X be a regular space, R a closed equivalence relation on X. Then the graph C of R in X × X is closed. ↩ 2 0B6W
Corollary On a regular space, every equivalence relation which is both open and closed is Hausdorff. 0B6X
Proposition 15 Let X be a regular space, F a closed subset of X, R the equivalence relation on X obtained by identifying all the points of F [in other words, the equivalence relation whose equiva… ↩ 2 0B6Y

§ 9. Compact spaces and locally compact spaces

Definition 1 A topological space X is said to be quasi-compact if it satisfies the following axiom: ↩ 1 0B7S
Example 1 Every finite space is quasi-compact, and more generally every space in which there is only a finite number of open sets is quasi-compact. A finite space is compact if and only if i… 0B7T
Example 2 Let X be a set, and give X the topology in which the closed sets are X and all finite subsets of X [this set of subsets clearly satisfies axioms (O'₁) and (O'_II) of § 1, no. 4]. T… ↩ 1 0B7U
Remark Quasi-compact (non-Hausdorff) spaces are of use mainly in applications of topology to algebraic geometry and are seldom featured in other mathematical theories, where on the contra… 0B7V
Theorem 1 Let 𝔉 be a filter on a quasi-compact space X and let A be the set of cluster points of 𝔉. Then every neighbourhood of A belongs to 𝔉. ↩ 3 0B7W
Corollary For a filter on a compact space to converge it is necessary and sufficient that it has a single cluster point. ↩ 3 0B7X
Proposition 1 Let X be a compact space, x a point of X. In order that a filter base B formed of closed neighbourhoods of x should be a fundamental system of neighbourhoods of x it is necessary a… ↩ 2 0B7Y
Corollary Every compact space is regular. ↩ 1 0B7Z
Proposition 2 Let X be a compact space and let A, B be two disjoint closed subsets of X. Then there exist two open sets U, V, such that U ∩ V = ∅ and A ⊂ U and B ⊂ V. ↩ 3 0B80
Definition 2 A subset A of a topological space X is said to be a quasi-compact (resp. compact) set if the subspace A is quasi-compact (resp. compact). 0B81
Example 1 In a topological space X, every finite subset is quasi-compact; the empty set and every set consisting of one point are compact. 0B82
Example 2 In a topological space X, let (xₙ)_n ∈ N be an infinite sequence of points which converges to a point a; then the set A consisting of the points xₙ (n ∈ N) and a is quasi-compact… ↩ 1 0B83
Proposition 3 Every closed subset of a quasi-compact (resp. compact) space is quasi-compact (resp. compact). ↩ 6 0B84
Proposition 4 Every compact subset of a Hausdorff space is closed. ↩ 8 0B85
Corollary In a compact space X a subset A is compact if and only if it is closed in X. 0B86
Proposition 5 The union of a finite family of quasi-compact subsets of a topological space is quasi-compact. ↩ 3 0B87
Definition 3 A subset A of a topological space X is said to be relatively quasi-compact (resp. relatively compact) in X if A is contained in a quasi-compact (resp. compact) subset of X. 0B88
Proposition 6 If X is a Hausdorff space, a subset A of X is relatively compact if and only if overlineA is compact. 0B89
Proposition 7 If A is a relatively quasi-compact subset of a topological space X, then every filter base on A has a cluster point in X. ↩ 1 0B8A
Remark In a non-Hausdorff space, a compact set need not be closed, and its closure need not be quasi-compact (Exercise 5); the intersection of two compact sets need not be quasi-compact (… 0B8B
Theorem 2 If f is a continuous mapping of a quasi-compact space X into a topological space X', then the set f(X) is quasi-compact. ↩ 17 0B8C
Corollary 1 Let f be a continuous mapping of a topological space X into a Hausdorff space X'. Then the image under f of any quasi-compact (resp. relatively quasi-compact) set in X is a compact… ↩ 9 0B8D
Corollary 2 Every continuous mapping f of a quasi-compact space X into a Hausdorff space X' is closed. If also f is bijective, then f is a homeomorphism. ↩ 9 0B8E
Corollary 3 A Hausdorff topology which is coarser than the topology of a quasi-compact space must coincide with the latter. ↩ 1 0B8F
Corollary 4 Let X be a topological space and R a Hausdorff equivalence relation on X. ↩ 2 0B8G
Theorem 3 (Tychonoff) Every product of quasi-compact (resp. compact) spaces is quasi-compact (resp. compact). Conversely, if a product of non-empty spaces is quasi-compact (resp. compact) then each of t… ↩ 6 0B8H
Corollary For a subset of a product of topological spaces to be relatively quasi-compact it is necessary and sufficient that each of its projections should be relatively quasi-compact in the… 0B8I
Proposition 8 Let (X_α, f_αβ) be an inverse system of compact spaces indexed by a directed set I such that f_αα is the identity mapping for each α ∈ I. Let X = varprojlim X_α be the inverse limi… ↩ 4 0B8J
Corollary 1 Let (X_α, f_αβ) be an inverse system of topological spaces indexed by a directed set, such that for each pair of indices α, β for which α ≤ β and for each x_α ∈ X_α, overlinef_αβ⁻¹… ↩ 2 0B8K
Corollary 2 Let (X_α, f_αβ) and (X'_α, f'_αβ) be two inverse systems of topological spaces indexed by the same directed set I, and let (u_α) be an inverse system of mappings u_α : X_α → X'_α… ↩ 2 0B8L
Definition 4 A topological space X is said to be locally compact if it is Hausdorff and if every point of X has a compact neighbourhood. 0B8M
Proposition 9 Every locally compact space is regular. 0B8N
Corollary In a locally compact space every point has a fundamental system of compact neighbourhoods. 0B8O
Proposition 10 In a locally compact space X, every compact set K has a fundamental system of compact neighbourhoods. ↩ 5 0B8P
Proposition 11 Let X be a locally compact space and F a subset of X such that F \cap K is compact whenever K is a compact subset of X. Then F is closed in X. 0B8Q
Proposition 12 In a Hausdorff space X, every locally compact subspace A is locally closed. ↩ 1 0B8R
Proposition 13 Every locally closed subspace of a locally compact space X is locally compact. ↩ 5 0B8S
Proposition 14 a) Let (Xᵢ)_i ∈ I be a family of locally compact spaces such that Xᵢ is compact for all but a finite number of indices. Then the product space X = ∏_i ∈ I Xᵢ is locally compact. ↩ 1 0B8T
Theorem 4 (Alexandroff) If X is any locally compact space, there exists a compact space X' and a homeomorphism f of X onto the complement of a point in X'. Furthermore, if X'₁ is another compact space suc… ↩ 8 0B8U
Definition 5 A locally compact space X is said to be σ-compact or countable at infinity if it is a countable union of compact subsets. 0B8V
Example 1 A discrete space is σ-compact if and only if it is countable. 0B8W
Example 2 \ The real line R is locally compact and σ-compact, since it is the union of the compact intervals [ -n, +n ] for n ∈ N. \ 0B8X
Remark A Hausdorff space can be a countable union of compact subspaces without being locally compact. \ An example is Hilbert space with the weak topology, as we shall show in a later vol… 0B8Y
Proposition 15 If X is a locally compact σ-compact space, there is a sequence (Uₙ) of relatively compact open subsets of X which cover X, such that overlineUₙ ⊂ Uₙ₊₁ for each n. ↩ 7 0B8Z
Corollary 1 With the notation of Proposition 15, every compact subset K of X is contained in some Uₙ. ↩ 3 0B90
Corollary 2 Let X be a locally compact space and let X' be the compact space obtained by adjoining a point at infinity ω to X (no. 8). Then X is σ-compact if and only if the point ω has a coun… ↩ 1 0B91
Definition 6 A topological space is said to be paracompact if it is Hausdorff and satisfies the following axiom: 0B92
Proposition 16 Every closed subspace F of a paracompact space X is paracompact. ↩ 1 0B93
Proposition 17 The product of a paracompact space and a compact space is paracompact. ↩ 2 0B94
Proposition 18 The sum (§ 2, no. 4, Example 3) of a family (Xᵢ)_i ∈ I of paracompact spaces is paracompact. ↩ 1 0B95
Theorem 5 A locally compact space X is paracompact if and only if X is the sum of a family of locally compact σ-compact spaces. ↩ 3 0B96
Corollary Let X be a locally compact paracompact space. Then every open covering ℜ of X has a locally finite open refinement ℜ' formed of relatively compact sets. If X is σ-compact then ℜ' c… 0B97

§ 10. Proper mappings

Example Every constant mapping into a Hausdorff space is closed. But it f is the constant mapping Q → 0, then f × ι_Q is the mapping (x, y) → (0, y) of Q² into Q², so it is the second proj… 0BA1
Definition 1 Let f be a mapping of a topological space X into a topological space Y. f is said to be proper if f is continuous and if the mapping f × ι_Z : X × Z → Y × Z is closed, for every to… ↩ 1 0BA2
Proposition 1 Every proper mapping is closed. ↩ 7 0BA3
Proposition 2 Let f : X → Y be a continuous injection. Then the following three statements are equivalent: a) f is proper. b) f is closed. c) f is a homeomorphism of X onto a closed subset of Y. ↩ 9 0BA4
Proposition 3 Let f : X → Y be a continuous mapping. If T is any subset of Y, let f_T denote the mapping overlinef⁻¹(T) → T which agrees with f on overlinef⁻¹(T). a) If f is proper, so is f_T. b… ↩ 8 0BA5
Proposition 4 Let I be a finite set and for each i ∈ I let fᵢ : Xᵢ → Yᵢ be a continuous mapping. Let X = ∏_i ∈ I Xᵢ, \ Y = ∏_i ∈ I Yᵢ, and let f : X → Y be the product mapping (xᵢ) → (fᵢ(xᵢ)). T… ↩ 4 0BA6
Proposition 5 Let f : X → X' and g : X' → X'' be two continuous mappings. a) If f and g are proper, then g ∘ f is proper. b) If g ∘ f is proper and f is surjective, then g is proper. c) If g ∘ f… ↩ 15 0BA7
Remark If X' is not Hausdorff it can happen that g ∘ f is proper and f not; for example, take X and X'' to consist of one point and X' to consist of two points, with the coarsest topology… 0BA8
Corollary 1 If f : X → Y is a proper mapping, then the restriction of f to a closed subset F of X is a proper mapping of F into Y. ↩ 1 0BA9
Corollary 2 Let f : X → Y be a proper mapping, where X is Hausdorff. Then the subspace f(X) of Y is Hausdorff. ↩ 2 0BAA
Corollary 3 Let I be a finite set and for each i ∈ I, let fᵢ : X → Yᵢ be a proper mapping. If X is Hausdorff, then the mapping x ↦ (fᵢ(x)) of X into ∏_i ∈ I Yᵢ is proper. ↩ 2 0BAB
Corollary 4 Let X and Y be two topological spaces, f : X → Y a continuous mapping, R the equivalence relation f(x) = f(y) on X, and X xrightarrowp X/R xrightarrowh f(X) xrightarrowi Y the cano… ↩ 1 0BAC
Lemma 1 Let X be a topological space such that the constant mapping X → P is proper. Then X is quasi-compact. ↩ 1 0BAD
Theorem 1 Let f : X → Y be a continuous mapping. Then the following four statements are equivalent: ↩ 19 0BAE
Lemma 2 If (fᵢ)_i ∈ I is a family of continuous mappings fᵢ : Xᵢ → Yᵢ, each of which satisfies condition d) of Theorem 1, then the product mapping f : (xᵢ) → (fᵢ(xᵢ)) also satisfies d). ↩ 1 0BAF
Corollary 1 A topological space X is quasi-compact if and only if the mapping X → P is proper. ↩ 4 0BAG
Corollary 2 Every continuous mapping f of a quasi-compact space X into a Hausdorff space Y is proper. ↩ 2 0BAH
Corollary 3 If (fᵢ) is a family of proper mappings, then the product mapping (xᵢ) → (fᵢ(xᵢ)) is proper. ↩ 2 0BAI
Corollary 4 Let X be a Hausdorff space, and let fᵢ : X → Yᵢ be a family of proper mappings. Then the mapping f : x → (fᵢ(x)) of X into ∏ᵢ Yᵢ is proper. 0BAJ
Corollary 5 If X is any quasi-compact space and Y is any topological space, then the projection pr₂ : X × Y → Y is proper. ↩ 7 0BAK
Example Let X be a set, and let f : X → X' be a mapping of X onto a topological space X'; topologize X with the inverse image under f of the topology of X'. Then f is proper, for f is clos… 0BAL
Remark When Y is Hausdorff, condition d) of Theorem 1 is equivalent to the following: 0BAM
Proposition 6 Let f : X → Y be a proper mapping, and let K be a quasi-compact subset of Y. Then overlinef⁻¹(K) is quasi-compact. ↩ 4 0BAN
Proposition 7 Let f be a continuous mapping of a Hausdorff space X into a locally compact space Y. Then f is proper if and only if the inverse image under f of every compact subset of Y is compa… ↩ 3 0BAO
Corollary Let X, X' be two locally compact spaces, and let Y (resp. Y') be the compact space obtained by adjoining a point at infinity ω (resp. ω') to X (resp. X') (§ 9, no. 8). Then a conti… ↩ 2 0BAP
Proposition 8 Let X be a compact space, R an equivalence relation on X, C the graph of R in X × X, f the canonical mapping X → X/R. Then the following conditions are equivalent: ↩ 8 0BAQ
Proposition 9 Let X be a locally compact space, R an equivalence relation on X, C the graph of R in X × X, f the canonical mapping X → X/R; let X' be the compact space obtained by adjoining a po… ↩ 2 0BAR
Corollary Let X be a Hausdorff space, Y a topological space, f : X → Y a proper mapping. Then for X to be compact (resp. locally compact) it is necessary and sufficient that f(X) is compact… 0BAS
Remark If X is locally compact but not compact, then a closed equivalence relation R on X need not be Hausdorff (Chapter IX, § 4, Exercise 14); and even if R is Hausdorff, X/R need not be… 0BAT
Proposition 10 Let X be a locally compact space, R an open Hausdorff equivalence relation on X, and let f : X \to X/R be the canonical mapping. Then X/R is locally compact, and if K' is any compa… ↩ 1 0BAU

§ 11. Connectedness

Definition 1 A topological space X is said to be connected if it is not the union of two disjoint non-empty open sets. 0BBG
Example 1 We shall see in Chapter IV, § 2, no. 5 that the real line is connected, and that the rational line is not. \ 0BBH
Example 2 A discrete space which has more than one point is not connected. 0BBI
Definition 2 A subset A of a topological space X is said to be a connected set if the subspace A of X is connected. 0BBJ
Example In any topological space, the empty set and every set consisting of a single point are connected. In a Hausdorff space X, every finite set consisting of more than one point is not… 0BBK
Proposition 1 If A is a connected set, then every set B such that A ⊂ B ⊂ overlineA is connected. ↩ 6 0BBL
Proposition 2 The union of a family of connected sets whose intersection is non-empty is connected. ↩ 4 0BBM
Corollary Let (Aₙ)_n ≥ 0 be an infinite sequence of connected sets such that Aₙ₊₁ ∩ Aₙ ≠ ∅ for all n ≥ 0. Then the union ⋃ₙ₌₀^∞ Aₙ is connected. 0BBN
Proposition 3 Let A be a subset of a topological space X. If B is a connected subset of X which meets both A and complement A, then B meets the frontier of A. ↩ 2 0BBO
Corollary In a connected space X, every non-empty set other than X itself has at least one frontier point. 0BBP
Proposition 4 Let A be a connected subset of a topological space X, and let f be a continuous mapping of X into a topological space X'. Then f(A) is connected. ↩ 11 0BBQ
Proposition 5 For a topological space X to be not connected it is necessary and sufficient that there exists a surjective continuous mapping of X onto a discrete space containing more than one p… ↩ 1 0BBR
Proposition 6 Every quotient space of a connected space is connected. ↩ 1 0BBS
Proposition 7 Let X be a topological space and R an equivalence relation on X. If the quotient space X/R is connected, and if each equivalence class mod R is connected, then X is connected. ↩ 1 0BBT
Proposition 8 Every product of connected spaces is connected. Conversely, if a product of non-empty spaces is connected, then each of the factors is connected. ↩ 5 0BBU
Definition 3 The component (or connected component) of a point of a topological space X is the largest connected subset of X which contains this point. The components of a subset A of X are the… ↩ 1 0BBV
Proposition 9 The component of any point in a topological space X is a closed set. The relation "y belongs to the component of x" is an equivalence relation R x, y on X, and the equivalence clas… ↩ 1 0BBW
Proposition 10 In a product space X = ∏_i ∈ J Xᵢ, the component of x = (xᵢ) in X is the product of the components of xᵢ in the factors Xᵢ. 0BBX
Definition 4 A topological space X is said to be locally connected if each point of X has a fundamental system of connected neighbourhoods. 0BBY
Proposition 11 A necessary and sufficient condition for a space X to be locally connected is that every component of an open set in X is open in X. ↩ 4 0BBZ
Corollary Let U be an open subset of a locally connected space X, and let V be a component of U. Then the frontier of V (relative to X) is contained in the frontier of U. 0BC0
Proposition 12 Every quotient space of a locally connected space is locally connected. ↩ 2 0BC1
Proposition 13 a) Let (Xᵢ)_i ∈ I be a family of locally connected spaces such that Xᵢ is connected for all but a finite number of indices i ∈ I. Then the product space X = ∏_i ∈ I Xᵢ is locally c… ↩ 1 0BC2
Theorem 1 Let X be a topological space satisfying axiom (O_III) (but not necessarily Hausdorff), and suppose X is connected and locally connected. Let Y be a topological space whose topology… ↩ 3 0BC3
Lemma 1 If x is any point of X, there is a distinguished pair (W, U) such that x ∈ W. ↩ 1 0BC4
Lemma 2 If (W, U) is a distinguished pair then the set of all distinguished pairs (W', U') such that W' meets W is countable. 0BC5
Corollary 1 Let Y be a regular space whose topology has a countable base (). Let X be a connected and locally connected space, and let p : X \to Y be a continuous mapping with the following pr… ↩ 2 0BC6
Corollary 2 Let X be locally compact, connected and locally connected, and suppose each point of X has a neighbourhood which has a countable base. Let Y be a Hausdorff space whose topology has… 0BC7
Corollary 3 (Poincaré-Volterra Theorem) Let Y be a locally compact, locally connected space whose topology has a countable base. Let X be a connected Hausdorff space, and let p : X → Y be a continuous mapping which has t… 0BC8

II. Uniform Structures

§ 1. Uniform spaces

Definition 1 A uniform structure (or uniformity) on a set X is a structure given by a set 𝔘 of subsets of X × X which satisfies axioms (F₁) and (F_II) of Chapter I, § 6, no. 1 and also satisfie… 0BCY
Remark 1 To make the language more expressive we may use the expressions "x is close enough to y" and "x and y are as close as we please" in some statements. For example, we shall say that… 0BCZ
Remark 2 The conjunction of axioms (U{II}) and (U{III}) is equivalent (assuming the other axioms of uniform structures) to the following axiom: 0BD0
Remark 3 If X is not empty, then axiom (UI) implies that no set of 𝔘 is empty, and therefore 𝔘 is a filter on X × X. There is only one uniformity on the empty set, namely 𝔘 = ∅ . 0BD1
Definition 2 A fundamental system of entourages of a uniformity is any set 𝔅 of entourages such that every entourage contains a set belonging to 𝔅. ↩ 1 0BD2
Proposition 1 Let X be a set endowed with a uniform structure U, and for each x ∈ X let 𝔅(x) be the set of subsets V(x) of X (*), where V runs through the set of entourages of U. Then there is a… ↩ 1 0BD3
Definition 3 The topology defined in Proposition 1 is called the topology induced by the uniform structure 𝔘. 0BD4
Example 1 The topology induced by the additive uniformity on the set of real numbers is the topology of the real line (Chapter I, § 1, no. 2); similarly the topology induced by the additive… 0BD5
Example 2 On any set X, the topology induced by the discrete uniformity (no. 1, Example 2) is the discrete topology. ↩ 2 0BD6
Proposition 2 Let X be a uniform space. For every symmetric entourage V of X and every subset M of X × X, VMV is a neighbourhood of M in the product space X × X, and the closure of M in this spa… ↩ 5 0BD7
Corollary 1 If A is any subset of X and V is any symmetric entourage of X, then V(A) is a neighbourhood of A in X, and ↩ 1 0BD8
Corollary 2 The interiors (resp. the closures) of the entourages of X in X × X form a fundamental system of entourages of X. ↩ 8 0BD9
Corollary 3 Every uniform space satisfies axiom (O_III). ↩ 2 0BDA
Proposition 3 A uniform space X is Hausdorff if and only if the intersection of all the entourages of its uniform structure is the diagonal Δ of X × X. Every Hausdorff uniform space is regular. ↩ 1 0BDB

§ 2. Uniformly continuous functions

Definition 1 A mapping f of a uniform space X into a uniform space X' is said to be uniformly continuous if, for each entourage V' of X', there is an entourage V of X such that the relation (x… ↩ 3 0BDH
Example 1 The identity mapping of a uniform space onto itself is uniformly continuous. 0BDI
Example 2 A constant mapping of a uniform space into a uniform space is uniformly continuous. 0BDJ
Example 3 Every mapping of a discrete uniform space into a uniform space is uniformly continuous. 0BDK
Proposition 1 Every uniformly continuous mapping is continuous. ↩ 1 0BDL
Proposition 2 (a) If f : X → X' and g : X' → X'' are two uniformly continuous mappings, then g ∘ f : X → X'' is uniformly continuous. ↩ 4 0BDM
Definition 2 If 𝔘₁ and 𝔘₂ are two uniform structures on the same set X, 𝔘₁ is said to be finer than 𝔘₂ (and 𝔘₂ coarser than 𝔘₁) if, denoting by Xᵢ the set X with the uniform structure 𝔘ᵢ (i = 1… 0BDN
Example In the ordered set of uniformities on a set X, the discrete uniformity is the finest, and the coarsest uniformity is that in which the set of entourages consists of the single elem… 0BDO
Proposition 3 If 𝔘₁ and 𝔘₂ are two uniformities on a set X, then 𝔘₁ is finer than 𝔘₂ if and only if every entourage of 𝔘₂ is an entourage of 𝔘₁. ↩ 1 0BDP
Corollary Let 𝔘₁ and 𝔘₂ be two uniformities on a set X, and suppose that 𝔘₁ is finer than 𝔘₂; then the topology induced by 𝔘₁ is finer than the topology induced by 𝔘₂. 0BDQ
Remark 1 It can happen that a uniformity 𝔘₁ is strictly finer than a uniformity 𝔘₂ but that the two induced topologies are identical. The following example shows this: ↩ 3 0BDR
Remark 2 If f : X → X' is a uniformly continuous mapping, then f remains uniformly continuous if we replace the uniformity of X by a finer uniformity and that of X' by a coarser uniformity… 0BDS
Proposition 4 Let X be a set, let (Yᵢ)_i ∈ I be a family of uniform spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. For each i ∈ I let gᵢ denote fᵢ × fᵢ. Let 𝔖 be the set of subsets… ↩ 6 0BDT
Corollary The topology on X induced by the coarsest uniformity 𝔘 for which the fᵢ are uniformly continuous is also the coarsest topology for which the fᵢ are continuous. ↩ 2 0BDU
Proposition 5 Let X be a set, let (Zᵢ)_i ∈ I be a family of uniform spaces, let (J_λ)_λ ∈ L be a partition of I and let (Y_λ)_λ ∈ L be a family of sets indexed by L. For each λ ∈ L, let h_λ be a… ↩ 4 0BDV
Remark If f : X → Y is surjective, then the entourages of Y are the direct images under g of the entourages of X. 0BDW
Definition 3 Let A be a subset of a uniform space X. The uniformity on A whose set of entourages is the trace on A × A of the set of entourages of X is called the uniformity induced on A by the… 0BDX
Proposition 6 Let A be a dense subset of a uniform space X. Then the closures, in X × X, of the entourages of the uniform subspace A form a fundamental system of entourages of X. ↩ 1 0BDY
Example If varpi is any finite partition (Aᵢ)_1 ≤ i ≤ n of a non-empty set X, the set V_varpi = ⋃ᵢ (Aᵢ × Aᵢ) by itself constitutes a fundamental system of entourages of a uniformity 𝔘_varp… 0BDZ
Remark A family (𝔘ᵢ) of uniformities on X also has a greatest lower bound in the ordered set of all uniformities on X, namely the least upper bound of the set of all uniformities on X whi… 0BE0
Definition 4 If (Xᵢ)_i ∈ I is a family of uniform spaces, the product uniform space of this family is the product set X = ∏_i ∈ I Xᵢ endowed with the coarsest uniformity for which the projectio… 0BE1
Proposition 7 Let f = (fᵢ) be a mapping of a uniform space Y into a product uniform space X = ∏_i ∈ I Xᵢ. Then f is uniformly continuous if and only if each fᵢ is uniformly continuous. ↩ 2 0BE2
Corollary Let (Xᵢ)_i ∈ I, (Yᵢ)_i ∈ I be two families of uniform spaces indexed by the same set I. For each i ∈ I, let fᵢ be a mapping of Xᵢ into Yᵢ. If each of the fᵢ is uniformly continuous… 0BE3
Proposition 8 Let X be a set, let (Yᵢ)_i ∈ I be a family of uniform spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let f be the mapping x → (fᵢ(x)) of X into Y = ∏_i ∈ I Yᵢ, and le… ↩ 2 0BE4
Corollary For each i ∈ I, let Aᵢ be a subspace of Yᵢ. Then the uniformity induced on A = ∏_i ∈ I Aᵢ by the product uniformity on ∏_i ∈ I Yᵢ is the same as the product of the uniformities of… 0BE5
Proposition 9 Let f be a uniformly continuous mapping of a product uniform space X₁ × X₂ into a uniform space Y; then every partial mapping ↩ 1 0BE6
Proposition 10 Let I be a directed set, let (X_α, f_αβ) be an inverse system of uniform spaces indexed by I, and let J be a cofinal subset of I. For each α ∈ I let f_α be the canonical mapping of… 0BE7

§ 3. Complete spaces

Definition 1 If X is a uniform space and if V is an entourage of X, a subset A of X is said to be V-small if every pair of points of A are V-close (in other words, if A × A ⊂ V). 0BEE
Proposition 1 In a uniform space X, if two sets A and B are V-small and intersect, then their union A ∪ B is vecV-small. ↩ 1 0BEF
Definition 2 A filter 𝔉 on a uniform space X is a Cauchy filter if for each entourage V of X there is a subset of X which is V-small and belongs to 𝔉. 0BEG
Proposition 2 On a uniform space X every convergent filter is a Cauchy filter. ↩ 1 0BEH
Proposition 3 Let f : X → X' be a uniformly continuous mapping. Then the image under f of a Cauchy filter base on X is a Cauchy filter base on X'. ↩ 3 0BEI
Proposition 4 Let X be a set, let (Yᵢ)_i ∈ I be a family of uniform spaces, and for each i ∈ I let fᵢ be a mapping of X into Yᵢ. Let X carry the coarsest uniformity U for which the fᵢ are unifor… ↩ 5 0BEJ
Corollary 1 If a Cauchy filter on a uniform space X induces a filter on a subset A of X, then this filter is a Cauchy filter on the uniform subspace A. 0BEK
Corollary 2 A filter base 𝔅 on a product ∏_i ∈ I Xᵢ of uniform spaces is a Cauchy filter base if and only if, for each i ∈ I, prᵢ(𝔅) is a Cauchy filter base on Xᵢ. ↩ 1 0BEL
Proposition 5 Let X be a uniform space. For each Cauchy filter 𝔉 on X there is a unique minimal Cauchy filter 𝔉₀ coarser than 𝔉. If 𝔅 is a base of 𝔉 and 𝔊 is a fundamental system of symmetric en… ↩ 8 0BEM
Corollary 1 For each x ∈ X, the neighbourhood filter 𝔅(x) of x in X is a minimal Cauchy filter. ↩ 1 0BEN
Corollary 2 Every cluster point x of a Cauchy filter 𝔉 is a limit point of 𝔉. ↩ 5 0BEO
Corollary 3 Every Cauchy filter, which is coarser than a filter converging to a point x, also converges to x. ↩ 2 0BEP
Corollary 4 If 𝔉 is a minimal Cauchy filter, then every set of 𝔉 has a non-empty interior which also belongs to 𝔉 (in other words, 𝔉 has a base consisting of open sets). ↩ 2 0BEQ
Example 1 Consider the sequence (uₙ) on the rational line Q defined by uₙ = ∑ₚ₌₀ⁿ 2^-p(p+1)/2. If m > n we have |uₘ - uₙ| ≤ 2^-n(n+3)/2 and therefore (uₙ) is a Cauchy sequence. But this sequ… 0BER
Example 2 Let X be an infinite set, and consider the uniformity of finite partitions on X (S 2, no. 2, Remark 1). Every ultrafilter 𝔉 on X is a 0BES
Definition 3 A complete space is a uniform space in which every Cauchy filter converges. 0BET
Example On a discrete uniform space X a Cauchy filter is a trivial ultrafilter (Chapter I, § 6, no. 4), hence convergent; consequently, X is complete. 0BEU
Proposition 6 Let 𝔉 be a filter on a set X, and let f be a mapping of X into a complete uniform space X'. Then f has a limit with respect to 𝔉 if and only if the image of 𝔉 under f is a Cauchy f… ↩ 1 0BEV
Proposition 7 Let 𝔘₁, 𝔘₂ be two uniformities on a set X, and let 𝔗₁, 𝔗₂ be the topologies induced by these uniformities respectively. Suppose that 𝔘₁ is finer than 𝔘₂, and that there is a fundam… ↩ 2 0BEW
Corollary In the conditions of Proposition 7, if U₂ is a uniformity of a complete space, then so is U₁. 0BEX
Proposition 8 Every closed subspace of a complete space is complete. Every complete subspace of a Hausdorff uniform space (complete or not) is closed. ↩ 6 0BEY
Proposition 9 Let X be a uniform space and let A be a dense subset of X such that every Cauchy filter base on A converges in X. Then X is complete. ↩ 2 0BEZ
Proposition 10 Every product of complete uniform spaces is complete. Conversely, if a product of non-empty uniform spaces is complete, then each of the factors is a complete uniform space. ↩ 5 0BF0
Corollary Let (X_α, f_αβ) be an inverse system of uniform spaces. If the X_α are Hausdorff and complete, then so is X = varprojlim X_α. ↩ 2 0BF1
Theorem 1 (Mittag-Leffler) Let (X_α, f_αβ) be an inverse system of complete Hausdorff uniform spaces, indexed by a directed set I which has a countable cofinal subset; suppose also that, for each α ∈ I, X_α… ↩ 3 0BF2
Corollary 1 Let (X_α, f_α β) be an inverse system of sets indexed by a directed set I which has a countable cofinal subset, and suppose that the f_α β are surjective. Then if X = varprojlim X_… ↩ 1 0BF3
Corollary 2 Let I be a directed set which has a countable cofinal subset. Let (X_α, f_α β) and (X'_α, f'_α β) be two inverse systems of sets indexed by I, and for each α ∈ I let u_α : X_α → X'… 0BF4
Example Suppose we are given in C: (i) a sequence (aₙ) of distinct points such that the sequence (|aₙ|) is increasing and tends to +∞; (ii) for each n, a rational function z → Rₙ(z) define… 0BF5
Proposition 11 Let A be a dense subset of a topological space X, and let f be a mapping of A into a complete Hausdorff uniform space X'. Then f can be extended by continuity to X if and only if… ↩ 3 0BF6
Theorem 2 Let f be a function defined on a dense subspace A of a uniform space X, taking its values in a complete Hausdorff uniform space X', and suppose that f is uniformly continuous on A… ↩ 7 0BF7
Corollary Let X₁, X₂ be two complete Hausdorff uniform spaces, and let Y₁, Y₂ be dense subspaces of X₁, X₂ respectively. Then every isomorphism f of Y₁ onto Y₂ extends to an isomorphism of X… ↩ 3 0BF8
Theorem 3 Let X be a uniform space. Then there exists a complete Hausdorff uniform space hatX and a uniformly continuous mapping i : X → hatX having the following property: ↩ 6 0BF9
Definition 4 The complete Hausdorff uniform space hatX defined in the proof of Theorem 3 is called the Hausdorff completion of X, and the mapping i : X → hatX is called the canonical mapping of… 0BFA
Proposition 12 (i) The subspace i(X) is dense in hatX. (ii) The graph of the equivalence relation i(x) = i(x') is the intersection of the entourages of X. (iii) The uniform structure of X is the… ↩ 5 0BFB
Corollary If X is a Hausdorff uniform space, then the canonical mapping i : X → hatX is an isomorphism of X onto a dense subspace of hatX. ↩ 2 0BFC
Remark If this identification is made, the minimal Cauchy filters on X are just the traces on X of the neighbourhood filters of points of hatX; this follows from the proof of Theorem 3. 0BFD
Proposition 13 If Y is a complete Hausdorff uniform space and X a dense subspace of Y, then the canonical injection X → Y extends to an isomorphism of hatX onto Y. 0BFE
Proposition 14 Let X be a complete Hausdorff uniform space, 𝔘 its uniformity, and let Z be a dense subspace of X. If 𝔘' is a uniformity on X which is coarser than 𝔘 and which induces the same uni… ↩ 1 0BFF
Proposition 15 Let X and X' be two uniform spaces. For each uniformly continuous mapping f : X → X' there is a unique uniformly continuous mapping hatf : hatX → hatX' such that the diagram ↩ 5 0BFG
Corollary If f : X → X' and g : X' → X'' are two uniformly continuous mappings and h = g ∘ f, then hath = hatg ∘ hatf. 0BFH
Proposition 16 Let X be a uniform space and i the canonical mapping of X into its Hausdorff completion hatX. For each uniformly continuous mapping f of X into a Hausdorff uniform space Y, there i… ↩ 2 0BFI
Definition 5 The Hausdorff uniform space i(X) defined in the proof of Theorem 3 is called the Hausdorff uniform space associated with X. 0BFJ
Corollary Let X, Y be two uniform spaces and X', Y' the associated Hausdorff spaces. For each uniformly continuous mapping f : X → Y there is a unique uniformly continuous mapping f' : X' →… ↩ 1 0BFK
Proposition 17 Let X be a uniform space, i(X) its associated Hausdorff space, and let f be a mapping of X onto a Hausdorff uniform space X', such that the uniformity of X is the inverse image und… ↩ 1 0BFL
Remark Let R be the equivalence relation i(x) = i(x') on X. We have seen (no. 7, Proposition 12) that the graph C of R is the intersection of all the entourages of X. It is clear that eve… 0BFM
Proposition 18 Let X be a set, let (Y_λ)_λ ∈ L be a family of uniform spaces, and for each λ ∈ L let f_λ be a mapping of X into Y_λ. Let X carry the coarsest uniformity U which makes all the f_λ… ↩ 2 0BFN
Corollary 1 Let X be a uniform space and let i be the canonical mapping of X into its Hausdorff completion hatX; let A be a subspace of X and j : A → X the canonical injection. Then hatj : hat… ↩ 3 0BFO
Corollary 2 Let (Y_λ)_λ ∈ L be a family of uniform spaces. Then the Hausdorff completion of the product space ∏_λ ∈ L Y_λ is canonically isomorphic to the product ∏_λ ∈ L hatY_λ. ↩ 1 0BFP

§ 4. Relations between uniform spaces and compact spaces

Definition 1 A uniformity on a topological space X is said to be compatible with the topology of X if the latter coincides with the topology induced by the uniformity. 0BFX
Theorem 1 On a compact space X there is exactly one uniformity compatible with the topology of X; the entourages of this uniformity are all the neighbourhoods of the diagonal Δ in X × X. Fur… ↩ 8 0BFY
Remark 1 For every finite open covering R = (Uᵢ)_1 ≤ i ≤ n of X, the set 0BFZ
Corollary 1 Every subspace of a compact space is uniformizable. 0BG0
Corollary 2 Every locally compact space is uniformizable. ↩ 1 0BG1
Remark 2 It can happen that there are several distinct uniformities compatible with the topology of a locally compact space. 0BG2
Theorem 2 Every continuous mapping f of a compact space X into a uniform space X' is uniformly continuous. ↩ 5 0BG3
Corollary Let A be a dense subspace of a compact space X, and let f be a mapping of A into a complete Hausdorff uniform space X'. Then f can be extended by continuity to the whole of X if an… 0BG4
Definition 2 A uniform space X is said to be precompact if its Hausdorff completion hatX is compact. A subset A of a uniform space X is said to be a precompact subset if the uniform subspace A… 0BG5
Example In any uniform space X, the set of points of a Cauchy sequence (xₙ) is precompact. Since the images of the xₙ in hatX again form a Cauchy sequence, we can assume that X is Hausdorf… 0BG6
Theorem 3 A uniform space X is precompact if and only if, for each entourage V of X, there is a finite covering of X by V-small sets. ↩ 3 0BG7
Corollary A uniform space X is compact if and only if it is Hausdorff and complete and can be covered by a finite number of V-small sets, where V is any entourage of X. ↩ 1 0BG8
Remark 1 A non-Hausdorff quasi-compact space is not necessarily uniformizable, since it need not satisfy axiom (O_III) (cf. Chapter I, § 9, no. 2); for example most of the non-Hausdorff qua… 0BG9
Proposition 1 In a uniform space every subset of a precompact set, every finite union of precompact sets and the closure of every precompact set are precompact. 0BGA
Remark 2 In a uniform space X a relatively compact set A is precompact, since A is contained in a compact set. On the other hand, even if X is Hausdorff, a precompact set need not be relati… 0BGB
Proposition 2 Let f : X → Y be a uniformly continuous mapping. If A is any precompact subset of X, then f(A) is a precompact subset of Y. ↩ 4 0BGC
Proposition 3 Let X be a set, let (Y_λ)_λ ∈ L be a family of uniform spaces, and for each λ ∈ L let f_λ be a mapping of X into Y_λ. Let X carry the coarsest uniformity for which the f_λ are unif… ↩ 5 0BGD
Proposition 4 In a uniform space X, let A be a compact set and B a closed set such that A ∩ B = ∅. Then there is an entourage V of X such that V(A) and V(B) do not intersect. ↩ 5 0BGE
Corollary Let A be a compact set in a uniform space X; then as V runs through the set of entourages of X, the sets V(A) form a fundamental system of neighbourhoods of A. ↩ 2 0BGF
Definition 3 Let V be a symmetric entourage of a uniform space X. A finite sequence (xᵢ)_0 ≤ i ≤ n of points of X is said to be a V-chain if xᵢ and xᵢ₊₁ are V-close for 0 ≤ i < n. The points x₀… 0BGG
Proposition 5 In a uniform space X, the set A_x, v of points which can be joined by a V-chain to a given point x is both open and closed in X. ↩ 3 0BGH
Proposition 6 In a compact space X, the component of x, the set Aₓ, and the intersection of the neighbourhoods of x which are both open and closed, are all three identical. ↩ 8 0BGI
Corollary Let X be a locally compact space and let K be a compact component of X. Then the neighbourhoods of K which are both open and closed form a fundamental system of neighbourhoods of K… ↩ 4 0BGJ
Proposition 7 Let X be a compact space and let R be the equivalence relation on X whose classes are the components of X. Then the quotient space X/R is compact and totally disconnected. 0BGK

III. Topological Groups

§ 1. Topologies on groups

Definition 1 A topological group is a set G which carries a group structure and a topology and satisfies the following two axioms: 0BH8
Example 1 The discrete topology on a group G is compatible with the group structure. A topological group whose topology is discrete is called a discrete group. 0BH9
Example 2 \ In Chapter IV we shall see that the topology of the rational line Q (resp. the real line R) is compatible with the additive group structure of Q (resp. R). 0BHA
Example 3 If G is a topological group, its topology is compatible with the structure of the group G⁰ which is the opposite of G; G⁰, with this topology, is said to be the topological group o… 0BHB
Proposition 1 Let G be a group and let 𝔅 be a filter on G satisfying the axioms (GV₁), (GV_II) and (GV_III). Then there is a unique topology on G, compatible with the group structure of G, for w… 0BHC
Remark If G is commutative, we have x . A . x⁻¹ = A for every subset A of G and every x ∈ G, and therefore (GV{III}) [resp. (GV'{III})] is automatically satisfied for every filter (resp… 0BHD
Proposition 2 A topological group G is Hausdorff if and only if the set e is closed. ↩ 3 0BHE
Corollary A topological group G is Hausdorff if and only if the intersection of the neighbourhoods of e consists only of the point e. ↩ 1 0BHF
Example Definition of a topology on a group by means of a set of subgroups. If 𝔅 is a filter base on a group G, formed of subgroups of G, then it is immediately seen that 𝔅 satisfies axiom… 0BHG
Definition 2 If G and G' are two topological groups, a local isomorphism of G with G' is a homeomorphism f of a neighbourhood V of the identity element of G onto a neighbourhood V' of the ident… ↩ 2 0BHH
Proposition 3 Let G and G' be two topological groups, and let f be a homeomorphism of a neighbourhood V of the identity element of G onto a neighbourhood V' of the identity element of G', which… ↩ 2 0BHI

§ 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups

Proposition 1 The closure overlineH of a subgroup H of a topological group G is a subgroup of G. If H is a normal subgroup of G, then so is overlineH. ↩ 4 0BHS
Proposition 2 If G is a Hausdorff topological group, then the closure of a commutative subgroup of G is a commutative subgroup of G. 0BHT
Proposition 3 Let G be a Hausdorff topological group and let M be any subset of G. Then the set M' of elements of G which commute with each element of M is a closed subgroup of G. In particular… 0BHU
Proposition 4 Let G be a topological group and let H be a subgroup of G which is locally closed at one point of H (Chapter I, § 3, no. 3, Definition 2). Then H is closed in G. ↩ 4 0BHV
Corollary A subgroup of a topological group is open if and only if it has an interior point. Every open subgroup is closed. ↩ 7 0BHW
Proposition 5 A subgroup H of a topological group G is discrete if and only if H has an isolated point. Every discrete subgroup of a Hausdorff group is closed. ↩ 1 0BHX
Remark Let H be any subgroup of a topological group G. For each x ∈ overlineH we have x overlineH = x. overlineH = overlineH, since translation by x is a homeomorphism of G onto G. In oth… 0BHY
Proposition 6 A connected topological group is generated by every neighbourhood of the identity element. ↩ 1 0BHZ
Proposition 7 In a topological group G, the component K of the identity element e is a closed normal subgroup. The component of any point x ∈ G is the coset x.K = K.x. ↩ 1 0BI0
Proposition 8 Let H be a dense subgroup of a topological group G, and let K be a normal subgroup of H. Then the closure overlineK of K in G is a normal subgroup of G. ↩ 3 0BI1
Proposition 9 Let H be a dense subgroup of a topological group G. If H is generated by every neighbourhood of the identity element in H, then G is generated by every neighbourhood of the identit… ↩ 1 0BI2
Lemma 1 If a topological group G operates continuously on a topological space X, then for each s ∈ G the mapping x → s.x is a homeomorphism of X onto X. ↩ 2 0BI3
Lemma 2 If a topological group G operates continuously on a topological space X, then the equivalence relation R defined by G is open. ↩ 10 0BI4
Example 1 Let H be a subgroup of a topological group G. H operates continuously on G by the external law (s, x) → sx. H also operates continuously on G by the external law (x, s) → sxs⁻¹. 0BI5
Example 2 \ If K is a topological division ring (§ 6, no. 7), the multiplicative group K^* operates continuously on K by the external law (s, x) → sx. \ 0BI6
Example 3 Let G be a topological group, X a topological space. Then the mapping (s, x) → x of G × X into X is an external law of composition on X, and G operates continuously on X with respe… 0BI7
Remark Instead of saying that a topological group G operates continuously on a topological space X, it is often said that G operates continuously on the left on X. When the topological gr… 0BI8
Proposition 10 The canonical bijection of the subspace φ(A) of X/G onto the orbit space A'/G is a homeomorphism. ↩ 1 0BI9
Proposition 11 If the equivalence relation S on X is open and compatible with G, then G operates continuously on X/S. ↩ 2 0BIA
Remark Let G' be another topological group operating continuously on X, and suppose that s.(s'x) = s'(s.x) for all s ∈ G, s' ∈ G' and x \in X. Then the equivalence relation S defined by G… 0BIB
Proposition 12 The group G operates continuously on every homogeneous space G/H. ↩ 1 0BIC
Proposition 13 Let G be a topological group and let H be a subgroup of G. Then the homogeneous space G/H is Hausdorff if and only if H is closed in G. ↩ 3 0BID
Proposition 14 Let G be a topological group and let H be a subgroup of G. Then the homogeneous space G/H is discrete if and only if H is open in G. 0BIE
Proposition 15 Let X be a topological space on which a topological group G operates continuously and transitively. For X to be a topological homogeneous space (relative to G) it is sufficient tha… ↩ 1 0BIF
Proposition 16 Let G be a topological group and let H be a normal subgroup of G. Then the quotient by H of the topology of G is compatible with the group structure of G/H. 0BIG
Proposition 17 Let φ be the canonical mapping of a topological group G onto a quotient group G/H. If 𝔅 is a fundamental system of neighbourhoods of e in G, then φ(𝔅) is a fundamental system of ne… ↩ 3 0BIH
Proposition 18 Let G be a topological group and let H be a normal subgroup of G. a) The quotient group G/H is Hausdorff if and only if H is closed in G. b) The quotient group G/H is discrete if a… ↩ 3 0BII
Proposition 19 If H is a discrete normal subgroup of a topological group G, then G/H is locally isomorphic to G. ↩ 3 0BIJ
Proposition 20 Let A be a subgroup of a topological group G, let H be a normal subgroup of G, and let φ denote the canonical mapping of G onto G/H. Then the canonical bijection of φ(A) onto AH/H… ↩ 4 0BIK
Proposition 21 Let G be a topological group, G₀ a dense subgroup of G, H₀ a normal subgroup of G₀, H the closure of H₀ in G and φ the canonical mapping G → G/H. Then the canonical bijection G₀/H₀… ↩ 1 0BIL
Proposition 22 Let G be a topological group operating continuously on a topological space X, and let H be a normal subgroup of G. Then the canonical bijection of X/G onto (X/H)/(G/H) is a homeomo… ↩ 2 0BIM
Corollary Let G be a topological group, H a normal subgroup of G, and K a normal subgroup of G which contains H. Then the canonical bijection of G/K onto (G/H)/(K/H) is an isomorphism of top… ↩ 2 0BIN
Proposition 23 A homomorphism f of a topological group G into a topological group G' is continuous on G if and only if it is continuous at one point of G. 0BIO
Definition 1 A continuous homomorphism of a topological group G into a topological group G' is said to be a strict morphism of G into G' if the bijective homomorphism dotf of G/overlinef⁻¹(e')… ↩ 1 0BIP
Proposition 24 Let f be a continuous homomorphism of a topological group G into a topological group G'. Then the following three statements are equivalent: a) f is a strict morphism. b) The image… ↩ 6 0BIQ
Remark 1 From condition b) of Proposition 24 it follows that every continuous homomorphism of a topological group into a discrete group is a strict morphism. ↩ 1 0BIR
Remark 2 Let f be a strict morphism of G into G' and let g be a strict morphism of G' into G''. If f is surjective or if g is injective, it follows immediately from Proposition 24 that g ∘… ↩ 1 0BIS
Remark 3 Let f be a continuous homomorphism of a topological group G into a topological group G', and let H be a normal subgroup of G. f induces a homomorphism g of the group G/H onto the q… ↩ 5 0BIT
Definition 2 The topological group obtained by giving the product set 0BIU
Proposition 25 Let (Gᵢ)_i ∈ I be a family of topological groups, and let H be the normal subgroup of G = ∏_i ∈ I Gᵢ consisting of all x = (xᵢ) such that the xᵢ are equal to the identity element e… ↩ 2 0BIV
Proposition 26 The bijection of X/G onto ∏_i ∈ I (Xᵢ/Gᵢ) canonically associated with (φᵢ) is a homeomorphism. For since the φᵢ are surjective and open, it follows that φ = (φᵢ) is an open mapping… 0BIW
Corollary Let (Gᵢ)_i ∈ I be a family of topological groups, and for each i ∈ I let Hᵢ be a normal subgroup of Gᵢ; let φᵢ denote the canonical mapping of Gᵢ onto Gᵢ/Hᵢ. Let G = ∏_i ∈ I Gᵢ, \… ↩ 3 0BIX
Remark If G is a commutative topological group, written additively, then the mapping (x, y) → x + y of G × G onto G is a strict morphism. For since (x + x') + (y + y') = (x + y) + (x' + y… 0BIY
Proposition 27 Let N and L be two groups, e' and e'' their respective identity elements. Suppose that we are given a homomorphism y → σ_y of L into the group Γ of automorphisms of N. Then: ↩ 1 0BIZ
Corollary The homomorphism h defined in 2) of Proposition 27 is injective if and only if f and g are injective and f(N) ∩ g(L) = e; and h is surjective if and only if f(N)g(L) = G. 0BJ0
Proposition 28 Let L, N be two topological groups, let y → σ_y be a homomorphism of L into the group of automorphisms Γ of the (non-topological) group structure of N; and suppose that the mapping… 0BJ1

§ 3. Uniform structures on groups

Definition 1 The right (resp. left) uniformity on a topological group G is the uniformity for which a fundamental system of entourages is obtained by making correspond to each neighbourhood V o… ↩ 1 0BJX
Proposition 1 The left and right translations are isomorphisms of the right uniformity onto itself. 0BJY
Proposition 2 The symmetry x → x⁻¹ is an isomorphism of the right uniformity onto the left uniformity. ↩ 3 0BJZ
Proposition 3 Every continuous homomorphism f of a topological group G into a topological group G' is uniformly continuous when considered as a mapping of G_d into G'_d (or of Gₛ into G'ₛ). ↩ 3 0BK0
Definition 2 A topological group is said to be complete if its left and right uniformities are structures of complete spaces. 0BK1
Proposition 4 If, in a topological group G, there is a neighbourhood V of e which is complete with respect to either the right or the left uniformity, then G is complete. ↩ 5 0BK2
Corollary 1 A locally compact group is complete. ↩ 2 0BK3
Corollary 2 Every locally compact subgroup of a Hausdorff topological group G is closed in G. 0BK4
Proposition 5 Let G₁ be a topological group, let G₂ be a complete Hausdorff topological group, and let H₁ (resp. H₂) be a dense subgroup of G₁ (resp. G₂). Then every continuous homomorphism u of… ↩ 7 0BK5
Remark If the continuous homomorphism u is bijective, it does not follow in general that baru is either injective or surjective (cf. Exercise 12); but see no. 5, Proposition 9. 0BK6
Proposition 6 Let 𝔉 and 𝔊 be two Cauchy filters on G_d. Then the image of the filter 𝔉 × 𝔊 under the mapping (x, y) → xy is a Cauchy filter base on G_d. 0BK7
Theorem 1 A Hausdorff topological group G is isomorphic to a dense subgroup of a complete group hatG if and only if the image, under the symmetry x → x⁻¹, of a Cauchy filter with respect to… ↩ 2 0BK8
Proposition 7 Let G be a Hausdorff topological group which has a completion hatG. Then the closures in hatG of the neighbourhoods of the identity element in G form a fundamental system of neighb… ↩ 2 0BK9
Proposition 8 Let G be a topological group which has a Hausdorff completion hatG'. Then every continuous homomorphism u of G into a complete Hausdorff group H can be uniquely factorized into u =… ↩ 2 0BKA
Theorem 2 Let G be a commutative topological group. Then the functions x⁻¹ and xy are uniformly continuous on G and G × G respectively. Moreover G admits a Hausdorff completion hatG, and hat… ↩ 2 0BKB
Proposition 9 Let G be a commutative group, and let 𝔗₁, 𝔗₂ be two Hausdorff topologies compatible with the group structure of G. Suppose that 𝔗₁ is finer than 𝔗₂ and that there is a fundamental… ↩ 5 0BKC
Corollary 1 Under the hypotheses of Proposition 9, if A is a subset of G which is a complete subspace with respect to the uniformity U₂ corresponding to 𝔗₂, then A is also a complete subspace… ↩ 1 0BKD
Corollary 2 Let G be a commutative group and let 𝔗₁, 𝔗₂ be two topologies compatible with the group structure of G. Suppose that 𝔗₁ is finer than 𝔗₂ and that there is a fundamental system 𝔅 of… ↩ 1 0BKE

§ 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators

Definition 1 Let G be a topological group operating continuously on a topological space X. G is said to operate properly on X if the mapping ↩ 1 0BKR
Example 1 Let H be a closed subgroup of a topological group G. If G operates properly on X, then so does H, since H × X is closed in G × X (Chapter I, § 10, no. 1, Proposition 5, Corollary 1… 0BKS
Example 2 If G operates properly on X, then it operates properly on every subspace X' of X which is a union of orbits of points of X (in other words, X' is saturated with respect to the equi… 0BKT
Proposition 1 Let G be a topological group operating continuously on a topological space X, and let K be a quasi-compact subset of G. Then the mapping ρ : (s, x) → s.x of K × X into X is proper. ↩ 7 0BKU
Corollary 1 If A is a closed (resp. compact) subset of X, then K.A is closed in X (resp. compact if X is Hausdorff). ↩ 9 0BKV
Corollary 2 If K is a quasi-compact subgroup of a topological group G, then the equivalence relation x⁻¹y ∈ K is closed, and the canonical mapping φ : G → G/K is proper. ↩ 1 0BKW
Corollary 3 Let K be a quasi-compact normal subgroup of a topological group G, and let φ be the canonical mapping G → G/K. Then for each closed subgroup A of G the canonical bijection of A/A ∩… ↩ 2 0BKX
Proposition 2 Let K be a compact group operating continuously on a Hausdorff space X. Then: ↩ 4 0BKY
Corollary 1 Under the hypotheses of Proposition 2, X is compact (resp. locally compact) if and only if X/K is compact (resp. locally compact). ↩ 1 0BKZ
Corollary 2 Let G be a Hausdorff topological group and let K be a compact subgroup of G. Then G is compact (resp. locally compact) if and only if G/K is compact (resp. locally compact). ↩ 3 0BL0
Proposition 3 If a topological group G operates properly on a topological space X, then the orbit space X/G is Hausdorff. If also G is Hausdorff, then X is Hausdorff. ↩ 4 0BL1
Proposition 4 Let G be a topological group operating properly on a topological space X, and let x be a point of X. Let G.x denote the orbit of x, and let Kₓ denote the stabilizer of x. Then: ↩ 6 0BL2
Remark Proposition 4 shows that if a topological group G operates properly on a homogeneous space G/H, then the subgroup H is quasi-compact (and therefore compact if G is Hausdorff). It c… 0BL3
Proposition 5 Let G (resp. G') be a topological group operating continuously on a topological space X (resp. X'). Let φ be a continuous homomorphism of G into G' and let ψ be a continuous mappin… ↩ 2 0BL4
Definition 2 Let G be a group operating on a set X. G is said to operate freely on X if the stabilizer of every element of X is e, in other words if the relations s.x = x,\ s ∈ G,\ x ∈ X imply… 0BL5
Example Let G be a group and let H be a subgroup of G. Then H operates freely by (left or right) translations on G. 0BL6
Proposition 6 Let G be a topological group operating continuously on a topological space X, and suppose that G operates freely on X. Then G operates properly on X if and only if the following co… 0BL7
Proposition 7 Let G be a locally compact group operating continuously on a Hausdorff space X. Then G operates properly on X if and only if, for each pair of points x, y of X, there is a neighbou… ↩ 5 0BL8
Proposition 8 Let G be a discrete group operating properly on a Hausdorff space X. Let x be a point of X and let Kx be the stabilizer of x. Then: ↩ 1 0BL9
Corollary If we suppose in addition that Kₓ = e , then the point x has an open neighbourhood U such that the restriction to U of the canonical mapping X → X/G is a homeomorphism of U onto an… 0BLA
Proposition 9 Let G be a topological group operating continuously on a locally compact space X. Then if X/G is Hausdorff it is locally compact. ↩ 3 0BLB
Proposition 10 Let G be a topological group operating continuously on a locally compact space X, and suppose that X/G is Hausdorff. Let φ be the canonical mapping of X onto X/G. Then if K' is any… ↩ 3 0BLC
Proposition 11 Let G be a Hausdorff topological group operating properly on a non-empty space X. If X is compact (resp. locally compact) then so are G and X/G. ↩ 2 0BLD
Theorem 1 Let G be a Hausdorff topological group which operates continuously on a topological space X. Let K be a compact subset of X and let L be a closed subset of X. Then: a) The set P(K… ↩ 5 0BLE
Remark Clearly, we have P(K, L) ⊂ P(K ∪ L, K ∪ L); therefore for G to operate properly on a locally compact space X, it is sufficient that, for each compact subset K of X, the set P(K, K)… 0BLF
Example Let X be a complex analytic manifold, analytically isomorphic to a bounded open subset of Cⁿ, and let G be the group of analytic automorphisms of X. The topology of compact converg… 0BLG
Proposition 12 Let G be a Hausdorff topological group which operates continuously on a topological space X. Let K be a compact subset of X, and let ρ_K be the mapping (s, x) → s.x of G × K into X… ↩ 2 0BLH
Corollary Let G be a Hausdorff topological group operating properly on a topological space X. If K is any compact subset of X and if F is any closed subset of G, then F.K is closed in X. ↩ 1 0BLI
Proposition 13 Let G be a locally compact group and let H be a closed subgroup of G. Then the homogeneous space G/H is locally compact and paracompact. ↩ 2 0BLJ
Proposition 14 In a locally compact group G, the identity component C is the intersection of the open subgroups of G. ↩ 3 0BLK
Corollary 1 If G is a totally disconnected locally compact group, then every neighbourhood of e in G contains an open subgroup of G. ↩ 3 0BLL
Corollary 2 A locally compact group is connected if it is generated by each neighbourhood of the identity element. 0BLM
Corollary 3 Let G be a locally compact group, let H be a closed subgroup of G, and let φ be the canonical mapping of G onto G/H. Then the components of G/H are the closures of the images under… ↩ 1 0BLN

§ 5. Infinite sums in commutative groups

Definition 1 Let (xᵢ)_i ∈ I be a family of points of a Hausdorff commutative group G; let 𝔉(I) be the set of finite subsets of the index set I, and for each finite subset J of I, let s_J be the… ↩ 3 0BMB
Remark 1 When I is finite, Definition 1 reduces to the ordinary definition of the sum of a finite family. More generally, if I is arbitrary and xᵢ = 0 except when the index i belongs to a f… 0BMC
Remark 2 The definition of a summable family does not involve any ordering of the index set I, and we may therefore say that the notion of a sum thus defined is commutative. More precisely… 0BMD
Remark 3 Definition 1 applies, more generally, to every family of points in a Hausdorff topological space X on which has been defined an associative and commutative law of composition, writ… ↩ 2 0BME
Theorem 1 (Cauchy's criterion) In a Hausdorff commutative group G, in order that a family (xᵢ)_i ∈ I should be summable it is necessary that, for each neighbourhood V of the origin in G, there is a finite subset… ↩ 6 0BMF
Proposition 1 If the family (xᵢ) is summable, then every neighbourhood of o contains all the xᵢ except for a finite subfamily (in other words, if I is infinite, we have lim xᵢ = o with respect t… ↩ 4 0BMG
Corollary Let (xᵢ)_i ∈ I be a summable family in a commutative group whose identity element has a countable fundamental system of neighbourhoods. Then the set of indices i such that xᵢ ≠ o i… ↩ 3 0BMH
Remark When G is written multiplicatively, Cauchy’s criterion takes the following form : for the family (xᵢ)_i ∈ I to be multipliable it is necessary that, for each neighbourhood V of the… 0BMI
Proposition 2 In a complete group G, every subfamily of a summable family is summable. ↩ 1 0BMJ
Theorem 2 (Associativity of the sum) Let (xᵢ)_i ∈ I be a summable family in a complete group G, and let (I_λ)_λ ∈ L be any partition of I. If s_λ denotes ∑_i ∈ I_λ xᵢ, then the family (s_λ)_λ ∈ L is summable and has t… ↩ 2 0BMK
Proposition 3 Let (xᵢ)_i ∈ I be a family of points of a group G, and let (I_λ)_λ ∈ L be a finite partition of I. If each of the subfamilies (xᵢ)_i ∈ I_λ is summable, then the family (xᵢ)_i ∈ I i… ↩ 2 0BML
Proposition 4 Let G = ∏_λ ∈ L G_λ be the product of a family of Hausdorff commutative groups. Then a family (xᵢ)_i ∈ I of points of G is summable if and only if, for each λ ∈ L, the family (pr_λ… ↩ 2 0BMM
Proposition 5 Let f be a continuous homomorphism of a commutative group G into a commutative group G'. If (xᵢ) is a summable family in G, then (f(xᵢ)) is a summable family in G', and we have 0BMN
Proposition 6 Let (xᵢ), (yᵢ) be two summable families, with the same index set, in a group G. Then the families (-xᵢ), (nxᵢ) (n ∈ Z), (xᵢ + yᵢ) are summable, and we have ↩ 1 0BMO
Remark Propositions 4 and 5 again apply to the case, mentioned earlier, of summable families in a topological space X which has an associative and commutative law of composition; the same… 0BMP
Proposition 7 If the series defined by the sequences (xₙ) and (yₙ) are convergent, then so are the series defined by the sequences (-xₙ) and (xₙ + yₙ), and we have (7) ∑ₙ₌₀^∞ (-xₙ) = -∑ₙ₌₀^∞ xₙ… ↩ 1 0BMQ
Corollary If (xₙ), (yₙ) are two sequences of points of G such that xₙ = yₙ except for a finite number of indices, and if the series whose general term is xₙ converges, then so does the serie… 0BMR
Proposition 8 (Restricted associativity of series) Let (kₙ) be a strictly increasing sequence of integers geqslant 0. If the series whose general term is xₙ converges, and if we put uₙ = ∑_p=kₙ₋₁^kₙ-1 xₚ, then the series whose gene… 0BMS
Definition 2 A series defined by a sequence (xₙ) is said to be commutatively convergent if, for each permutation σ of N, the series defined by the sequence (x_φ(n)) is convergent. 0BMT
Proposition 9 The series defined by the sequence (xₙ) is commutatively convergent if and only if the sequence (xₙ) is summable; and then, for each permutation σ of N, we have ↩ 6 0BMU

§ 6. Topological groups with operators; topological rings division rings and fields

Proposition 1 If H is a stable subgroup of a topological group G with operators, then the closure overlineH of H in G is a stable subgroup of G. ↩ 1 0BN2
Definition 1 Let E be a commutative topological group with operators, and let (Mᵢ)_1 ≤ i ≤ n be a finite family of stable subgroups of E, such that E is the direct sum of the Mᵢ. Then E is said… 0BN3
Proposition 2 Let E be a commutative topological group with operators which is the direct sum of stable subgroups Mᵢ (1 ≤ i ≤ n). Let (pᵢ)_1 ≤ i ≤ n be the family of projectors associated with t… ↩ 1 0BN4
Corollary Let E be a commutative topological group with operators, and let M be a stable subgroup of E. Then the following conditions are equivalent: ↩ 2 0BN5
Remark 1 To avoid confusion, we shall sometimes say that a stable subgroup of E which is a complement of M (in the sense of the structure of group with operators, without a topology) is an… 0BN6
Remark 2 If a Hausdorff commutative topological group with operators is the topological direct sum of a family (Mᵢ)_1 ≤ i ≤ n of stable subgroups, then each of the subgroups Mᵢ is closed in… 0BN7
Proposition 3 Let E, F be two commutative topological groups with operators, and let u be a continuous linear mapping of E into F. In order that there should exist a continuous linear mapping v… ↩ 1 0BN8
Proposition 4 Let E, F be two commutative topological groups with operators, and let u be a continuous linear mapping of E into F. In order that there should exist a continuous linear mapping v… 0BN9
Definition 2 A topological ring is a set A which carries a ring structure and a topology satisfying the following axioms: 0BNA
Example 1 On any ring A, the discrete topology is compatible with the ring structure. A topological ring whose topology is discrete is called a discrete ring. 0BNB
Example 2 We shall see in Chapter IV that the topology of the rational line Q (resp. the real line R) is compatible with the ring structure of Q (resp. R). 0BNC
Remark In analysis one fairly often meets rings which satisfy axioms (ATI), (AT{II}) and (AT{III a}), but not (AT{III b}). An example is the ring of measures on a compact group, where mul… 0BND
Proposition 5 Let H be a dense subring of a topological ring A, and let K be a subring (resp. left ideal, right ideal, two-sided ideal) of H. Then the closure overlineK of K in A is a subring (r… ↩ 2 0BNE
Theorem 1 Let E, F, G be three complete Hausdorff commutative groups; let A be a dense subgroup of E and let B be a dense subgroup of F. If f is a Z continuous Z-bilinear () mapping of A × B… ↩ 2 0BNF
Proposition 6 A Hausdorff topological ring A is isomorphic to a dense subring of a complete Hausdorff ring hatA, which is determined up to isomorphism (and is called the completion of A). ↩ 2 0BNG
Definition 3 Given a topological ring A with an identity element, a topological left A-module is a set E, together with: 0BNH
Example 1 A topological vector space over R (resp. C) is a topological R (resp. C) module. 0BNI
Example 2 Let A be a ring and let 𝔅 be a filter base on A consisting of two-sided ideals of A; let E be a left A-module. If we give A the topology (compatible with its ring structure) for wh… 0BNJ
Remark Given a topological ring A, consider, on a left A-module E, a topology compatible with the additive group structure of E. By virtue of the identity λ x - λ₀ x₀ = (λ - λ₀)x₀ + λ₀(x… 0BNK
Definition 4 A topological division ring is a set K carrying a division ring structure and a topology compatible with the ring structure of K, and satisfying in addition the following axiom: ↩ 1 0BNL
Example 1 On any division ring K, the discrete topology is compatible with the division ring structure. A topological division ring whose topology is discrete is called a discrete division r… 0BNM
Example 2 \ The topology of the rational line Q (resp. the real line R) is compatible with the field structure of R (resp. R) (see Chapter IV, § 3). \ 0BNN
Proposition 7 The completion hatK of a Hausdorff topological division ring K is a topological division ring if and only if the image under the mapping x → x⁻¹ of every Cauchy filter (with respec… ↩ 2 0BNO
Proposition 8 If the additive uniform structure of a topological field K is a structure of a complete Hausdorff space, then the multiplicative structure on K is a structure of a complete space. ↩ 3 0BNP

§ 7. Inverse limits of topological groups and rings

Proposition 1 Suppose that the X_α and the G_α satisfy the above hypotheses. ↩ 1 0BOG
Corollary 1 If the G_α are compact and the X_α Hausdorff, then the conclusions of a) and b) are valid. ↩ 1 0BOH
Corollary 2 If, for each α ∈ I, the group G_α operates transitively on the space X_α, and if the stabilizer of each point of X_α is a compact subgroup of G_α, then G operates transitively on X… 0BOI
Corollary 3 Suppose that the G_α are Hausdorff. For each α ∈ I, let K_α be a compact subgroup of G_α, such that f_α β (K_β) ⊂ K_α whenever α ≤ β. Then, if K = varprojlim K_α, the canonical map… 0BOJ
Proposition 2 Let G be a topological group and let (H_α)_α ∈ I be a family of normal subgroups of G such that H_α ⊃ H_β whenever α ≤ β, and which satisfy the following condition: ↩ 8 0BOK
Corollary 1 If the condition (AP) is satisfied and if in addition the (Hausdorff) groups G/H_α are complete, then the group G has a Hausdorff completion which can be identified with tildeG; an… ↩ 1 0BOL
Corollary 2 Let G be a group and let (H_α) be a family of normal subgroups of G, directed with respect to the relation H_α ⊃ H_β. If we endow G with the group topology for which the H_α form a… 0BOM
Lemma 1 Under the hypotheses of Proposition 2, if E is any closed subset of G we have E = ⋂_α E H_α. ↩ 1 0BON
Proposition 3 Suppose that G is Hausdorff and that the H_α are compact and satisfy (AP). ↩ 3 0BOO
Proposition 4 Suppose that G is Hausdorff, the H_α are compact and that (AP) is satisfied. If C_α is the identity component of G_α = G/H_α, then the identity component C of G can be identified w… ↩ 1 0BOP
Lemma 2 Let G be a Hausdorff topological group and let H be a compact normal subgroup of G, and φ the canonical mapping G → G/H. If C is the identity component of G, then φ(C) is the ident… 0BOQ
Remark If the subgroup H_α is compact for some α ∈ I, then H_β is compact for all β ≥ α, because it is a closed subgroup of H_α. Since the set of all β ∈ I such that β ≥ α is cofinal in I… 0BOR
Proposition 5 Let (G_α, f_αβ) be an inverse system of Hausdorff topological groups such that the f_αβ are surjective strict morphisms with compact kernels. Then for each α ∈ I the canonical mapp… ↩ 3 0BOS
Corollary 1 Let (G_α, f_α β) be an inverse system of topological groups which satisfies the hypotheses of Proposition 5; let (G'_α, f'_α β) be an inverse system of topological groups, and for… 0BOT
Corollary 2 Let (G_α, f_α β) be an inverse system of topological groups which satisfies the conditions of Proposition 5, and let G' be a topological group in which there is a neighbourhood V'… 0BOU

IV. Real Numbers

§ 1. Definition of real numbers

Definition 1 The rational line is the topological space consisting of the set Q together with the additive group topology for which the symmetric open intervals ]-a, +a[ (a > 0) form a fundamen… 0BP1
Definition 2 Let R denote the topological group which is the completion of the additive group Q of the rational line. The elements of R are called real numbers; as a topological space, R is cal… 0BP2
Proposition 1 The relation y - x ∈ overlineQ₊ is an ordering on R which makes R into a linearly ordered set; is compatible with the additive group structure on R, and induces the ordering x ≤ y… 0BP3
Proposition 2 Every closed (resp. open) interval in R is a closed (resp. open) set in R. ↩ 1 0BP4
Proposition 3 As r runs through the set of rational numbers > 0, the intervals sᵣ = [-r, +r] in R form a fundamental system of neighbourhoods of 0. ↩ 2 0BP5
Corollary Every point of the real line has a countable fundamental system of neighbourhoods. ↩ 1 0BP6
Proposition 4 If (x, y) is any pair of real numbers such that x < y, there is a rational number r such that x < r < y. 0BP7
Proposition 5 Let I be any interval in \mathbf{R}. Then the topology induced on I by the topology of \mathbf{R} is generated by the open intervals of I (where I is considered as linearly ordered… ↩ 1 0BP8
Remark If A is a dense subset of \mathbf{R}, the topology of \mathbf{R} is generated by the open intervals whose end-points belong to A. For if 0BP9
Definition 3 The length of a bounded interval with end-points a and b (a \leq b) is defined to be b - a. 0BPA

§ 2. Fundamental topological properties of the real line

Theorem 1 If x and y are any two real numbers 0, then there is an integer n > 0 such that y < nx. ↩ 3 0BPF
Remark An axiomatic construction of the theory of real numbers will be found in Chapter V, S 2, in which Theorem 1 appears as an axiom; for more details about this axiom, see the Historic… 0BPG
Theorem 2 (Borel-Lebesgue) For a subset of the real line R to be compact it is necessary and sufficient that it be closed and bounded. ↩ 4 0BPH
Corollary 1 A subset of the real line R is relatively compact if and only if it is bounded. 0BPI
Corollary 2 The real line is a locally compact space and is not compact. 0BPJ
Remark Theorem 2 is often referred to as the "Heine-Borel Theorem"; see the Historical Notes to Chapters II and IV. 0BPK
Theorem 3 Every non-empty subset of the real line which is bounded above (resp. bounded below) has a least upper bound (resp. greatest lower bound). ↩ 1 0BPL
Proposition 1 A non-empty subset A of R is an interval if and only if, whenever a and b are any two points of A such that a < b, the closed interval [a, b] is contained in A. ↩ 3 0BPM
Theorem 4 A subset A of R is connected if and only if A is an interval. ↩ 4 0BPN
Corollary 1 The real line is a connected and locally connected space. 0BPO
Corollary 2 The only compact connected subsets of R are the bounded closed intervals. 0BPP
Proposition 2 Every non-empty open set in R is the union of a countable family of mutually disjoint open intervals. ↩ 2 0BPQ
Example Let us define by induction a countable family (I_n,p) of mutually disjoint open intervals as follows: The integer n takes all values geqslant 0, and for each value of n, p takes th… 0BPR
Theorem 5 Let I be an interval in R. Then a mapping f of I into R is a homeomorphism of I onto f(I) if and only if f is strictly monotonic and continuous on I; and f(I) is then an interval i… ↩ 4 0BPS
Remark The first part of the preceding proof shows in fact that a continuous injective mapping of I into R is strictly monotonic; from the second part of the proof it therefore follows th… 0BPT

§ 3. The field of real numbers

Proposition 1 The functions xy and 1/x, defined respectively on Q × Q and Q^*, can be extended by continuity to R × R and R^* respectively, and define a field structure on R. Endowed with this s… 0BQF
Proposition 2 The multiplicative group R^* of real numbers ≠ 0 is a topological group isomorphic to the product of its subgroups R₊^* and U₀, where 0BQG

§ 4. The extended real line

Proposition 1 All non-empty open intervals of R are homeomorphic to R. ↩ 4 0BQK
Remark To show that all bounded open intervals are homeomorphic to each other, it would be enough to remark that, if a ≠ b and a' ≠ b', there is a homeomorphism of R onto itself, of the f… 0BQL
Definition 1 The set overlineR endowed with the order structure and the topology defined above is called the extended real line. ↩ 1 0BQM
Proposition 2 Every homeomorphism f of R onto an interval ]a, b[ can be extended to a homeomorphism overlinef of overlineR onto [a, b]. If f is an increasing function, then overlinef is an order… ↩ 2 0BQN
Proposition 3 The extended real line is compact. 0BQO
Proposition 4 Every non-empty subset of overlineR has a least upper bound and a greatest lower bound. 0BQP
Proposition 5 A subset A of overlineR is connected if and only if A is an interval. ↩ 1 0BQQ
Corollary The extended real line is a connected, locally connected space. 0BQR
Proposition 6 A mapping f of an interval I of overlineR into overlineR is a homeomorphism of I onto f(I) if and only if f is strictly monotonic and continuous on I; f(I) is then an interval of o… 0BQS
Proposition 7 The function x + y can be extended by continuity to each of the sets A' × A' and A'' × A'', according to the formulae ↩ 1 0BQT
Proposition 8 The function xy can be extended by continuity to the set overlineR^* × overlineR^* according to the formulae 0BQU

§ 5. Real-valued functions

Definition 1 A mapping of a set X into the real line is called a real-valued function (or real function) defined on X. 0BR2
Proposition 1 Let f and g be two real-valued functions defined on a set X filtered by a filter 𝔉. If lim_𝔉 f and lim_𝔉 g exist, and if, for each subset A ∈ 𝔉, there exists x ∈ A such that f(x) ≤… ↩ 1 0BR3
Proposition 2 Let f and g be two real-valued functions defined on a set X filtered by a filter 𝔉. If lim_𝔉 f and lim_𝔉 g exist, and if lim_𝔉 f > lim_𝔉 g, then there is a set A ∈ 𝔉 such that f(x)… 0BR4
Theorem 1 (Principle of extension of inequalities) Let f and g be two real-valued functions, defined on a set X filtered by a filter 𝔉. If lim_𝔉 f and lim_𝔉 g exist, and if f ≤ g, then lim_𝔉 f ≤ lim_𝔉 g. ↩ 6 0BR5
Remark If in particular we have f(x) < g(x) for all x ∈ X (or only for all points of a set of the filter overlineR) we can infer, by Theorem 1, that lim_𝔉 f ≤ lim_𝔉 g; but we cannot infer… 0BR6
Theorem 2 (Theorem of the monotone limit) Let X be an ordered set and let A be a directed subset of X (). Every monotonic real-valued function f defined on A has a limit with respect to A (Chapter I, , no. 3); if f is incr… ↩ 6 0BR7
Corollary An increasing (resp. decreasing) real-valued function, defined on a directed subset A of an ordered set X, has a finite limit with respect to A if and only if it is bounded above (… ↩ 1 0BR8
Proposition 3 Every monotonic sequence of real numbers has a limit in overlineR. In particular, every increasing (resp. decreasing) sequence of finite numbers converges to a finite real number i… 0BR9
Definition 2 Let f be a function defined on a non-empty subset A of overlineR, with values in a topological space X. A limit of f with respect to the filter 𝔉, if it exists, is called a limit o… 0BRA
Proposition 4 Let A be a subset of overlineR and let a ≠ -∞ be a point in the closure of the intersection A ∩ [-∞, a[. If f is a monotonic real-valued function defined on A, then f has a limit f… 0BRB
Definition 3 Let f be a real-valued function defined on a set X, and let A be a non-empty subset of X. Then the least upper bound (resp. greatest lower bound) of the set f(A) in overlineR is ca… 0BRC
Proposition 5 Let f be a real-valued function defined on a set X. On the set 𝔉(X) of all finite subsets of X, directed with respect to the relation ⊂, the real-valued function H → sup_x ∈ H f(x)… ↩ 2 0BRD
Proposition 6 Let f and g be two real-valued functions defined on X. If f(x) ≤ g(x) at every point x of a non-empty subset A of X, then we have 0BRE
Proposition 7 Let f be a real-valued function defined on X. If A and B are two non-empty subsets of X such that A ⊂ B, then ↩ 1 0BRF
Proposition 8 Let f be a real-valued function defined on X, and let (Aᵢ)_i ∈ I be a non-empty family of non-empty subsets of X; then 0BRG
Proposition 9 Let f be a real-valued function defined on a product set X₁ × X₂. If A₁, A₂ are any non-empty subsets of X₁, X₂ respectively, then ↩ 2 0BRH
Definition 4 Let (fᵢ)_i ∈ I be a family of real-valued functions defined on a set X. The real-valued function on X whose value at each point x ∈ X is sup_i ∈ I (fᵢ(x)) [resp. inf_i ∈ I (fᵢ(x))]… 0BRI
Proposition 10 In the product space overlineR^X the upper envelope sup_i ∈ I fᵢ of a family of real-valued functions (fᵢ)_i ∈ I is the limit, with respect to the directed set 𝔉(I) of finite subse… 0BRJ
Definition 5 A family (fᵢ)_i ∈ I of real-valued functions defined on a set X is said to be uniformly bounded above (resp. uniformly bounded below) in X, if there exists a finite number a such t… 0BRK
Definition 6 The limit of the real-valued function M → sup_x ∈ M f(x) with respect to the directed set 𝔊 is called the upper limit of f with respect to the filter 𝔊, and is denoted by lim sup_𝔊… 0BRL
Theorem 3 The upper limit of a real-valued function f with respect to a filter 𝔊 is equal to the largest cluster value of f with respect to 𝔊. 0BRM
Corollary 1 In order that lim sup_𝔊 f = lim inf_𝔊 f, it is necessary and sufficient that f has a limit with respect to the filter 𝔊, and then lim_𝔊 f = lim sup_𝔊 f = lim inf_𝔊 f. For since ove… ↩ 1 0BRN
Corollary 2 If ℌ is a filter finer than 𝔊, we have lim inf_𝔊 f ≤ lim inf_ℌ f ≤ lim sup_ℌ f ≤ lim sup_𝔊 f. For every cluster point of f with respect to ℌ is also a cluster point of f with respe… ↩ 1 0BRO
Corollary 3 Let A be a set of the filter 𝔊, let 𝔊_A be the filter induced on A by 𝔊, and let f_A be the restriction of f to A; then lim sup_𝔊_A f_A = lim sup_𝔊 f. For every cluster point of th… ↩ 1 0BRP
Proposition 11 Let f and g be two real-valued functions defined on a filtered set X. Then the relation f ≤ g implies 0BRQ
Proposition 12 Let f and g be two real-valued functions defined on a set X, and let A be a non-empty subset of X. ↩ 1 0BRR
Corollary 1 Let f be a real-valued function defined on X, and let k be a real number. Then (22) sup_x ∈ A (f(x) + k) = k + sup_x ∈ A f(x) whenever both sides are defined, and, if k ≥ 0, (23) s… 0BRS
Corollary 2 Let f₁ and f₂ be two real-valued functions defined on sets X₁, X₂ respectively; then if A₁, A₂ are any non-empty subsets of X₁, X₂ respectively, we have (24) sup_(x₁, x₂) ∈ A₁ × A₂… 0BRT
Proposition 13 Let f and g be two real-valued functions defined on a filtered set X. 0BRU
Corollary 1 Let f and g be two real-valued functions defined on a filtered set X. If lim g exists, then 0BRV
Corollary 2 Let f and g be two real-valued functions defined on a filtered set X. If lim f = +∞ and lim inf g > -∞ and f + g is defined, then lim (f + g) = +∞. If lim f = +∞ and lim inf g > 0… 0BRW

§ 6. Continuous and semi-continuous real-valued functions

Theorem 1 (Weierstrass) Let f be a continuous real-valued function defined on a non-empty quasi-compact space X. Then there is at least one point a ∈ X such that f(a) = sup_x ∈ X (f(x)), and at least one… ↩ 5 0BSF
Corollary If a real-valued function defined on a non-empty quasi-compact space X is continuous and finite on X, then it is bounded in X. 0BSG
Theorem 2 (Bolzano) Let f be a continuous real-valued function defined on a connected space X. If a and b are any two points of X, and if α is a real number belonging to the closed interval whose end-… ↩ 1 0BSH
Definition 1 A real-valued function f, defined on a topological space X, is said to be lower semi-continuous (resp. upper semi-continuous) at a point a ∈ X, if for each h < f(a) [resp. each k >… ↩ 1 0BSI
Example 1 If f has a relative minimum at a point a, that is to say if there is a neighbourhood V of a such that, for each x ∈ V, we have f(a) ≤ f(x), then f is lower semi-continuous at a. In… 0BSJ
Example 2 Define a real-valued function f on R by putting f(x) = 0 if x is irrational, and f(x) = 1/q if x is rational and equal to the irreducible fraction p/q (q > 0). For each integer n >… 0BSK
Proposition 1 A real-valued function f on a topological space X is lower semi-continuous if and only if, for each finite real number k, overlinef([k, +∞]) [the set of all x ∈ X such that f(x) >… ↩ 3 0BSL
Corollary A subset A of a topological space X is open (resp. closed) in X if and only if its characteristic function () φ_A is lower (resp. upper) semi-continuous on X. ↩ 1 0BSM
Theorem 3 Let f be a lower semi-continuous function on a non-empty quasi-compact space X. Then there is at least one point a ∈ E such that f(a) = inf_x ∈ X f(x) (in other words, f attains it… ↩ 1 0BSN
Corollary Let f be a lower semi-continuous function on a non-empty quasi-compact space X. If f(x) > -∞ for all x ∈ X, then f is bounded below in X. 0BSO
Proposition 2 Let f and g be two real-valued functions, lower semi-continuous at a point a ∈ X. Then the functions inf(f, g) and sup(f, g) are lower semi-continuous at a; so is f + g whenever it… ↩ 1 0BSP
Theorem 4 The upper envelope of a family (fᵢ) of functions which are lower semi-continuous at a point a ∈ X is lower semi-continuous at a. ↩ 2 0BSQ
Corollary The upper envelope of a family of continuous real-valued functions on a space X is lower semi-continuous on X. ↩ 1 0BSR
Proposition 3 A real-valued function f, defined on a topological space X, is lower semi-continuous at a point a ∈ X if and only if liminf_x → a f(x) = f(a) [or, equivalently, if and only if limi… ↩ 1 0BSS
Proposition 4 Let f be any real-valued function defined on a dense subset A of a topological space X. If, for each x ∈ X, we put g(x) = liminf_y → x, y ∈ A f(y), then g is lower semi-continuous… 0BST

§ 7. Infinite sums and products of real numbers

Theorem 1 A family (xᵢ) of finite real numbers geqslant 0 is summable in R if and only if the set of partial finite sums of the family is bounded above in R. If so, the least upper bound of… ↩ 5 0BT9
Remark Let (H_λ) be a family of finite subsets of I such that, for each finite subset H of I, there is an index λ such that H ⊂ H_λ; then (xᵢ) is summable if and only if the family of the… 0BTA
Examples 1) For each number q such that 0 leqslant q < 1, the sequence (qⁿ) ("geometric progression of ratio q") is summable in R, since sₙ = frac1 - qⁿ⁺¹1 - q leqslant 1/(1 - q) the sum of… 0EAD
Theorem 2 (Principle of comparison) Let (xᵢ)_i ∈ I and (yᵢ)_i ∈ I be two families of finite real numbers ≥ 0, such that xᵢ ≤ yᵢ for all i. If (yᵢ) is summable in R then so is (xᵢ), and we have ∑ᵢ xᵢ ≤ ∑ᵢ yᵢ; if in ad… ↩ 2 0BTB
Example 1 Let a be a finite real number > 0, and consider the sequence ( aⁿ/n! ); let n₀ be the smallest integer such that a < n₀. Then for each n ≥ n₀ we have aⁿ/n! ≤ fraca^n₀n₀! · ( a/n₀ )… 0BTC
Example 2 Let (aₙ) be a summable sequence of positive numbers. Since lim_n → ∞ aₙ = 0, there exists an integer n₀ such that aₙ ≤ 1 whenever n ≥ n₀. Consequently, for each n ≥ n₀ we have aₙ²… 0BTD
Example 3 Let a and b be two numbers < 1; then 1/(aᵐ + bⁿ) ≤ frac12(sqrta)ᵐ (sqrtb)ⁿ and hence the family ( 1/(aᵐ + bⁿ) ) is summable in R. 0BTE
Corollary Let (xᵢ)_i ∈ I be a summable family of finite numbers ≥ 0 in R. If H is any subset of I, we have ∑_i ∈ H xᵢ ≤ ∑_i ∈ I xᵢ, and equality holds only if xᵢ = 0 for all i ∈ complement H… 0BTF
Theorem 3 Let (xᵢ)_i ∈ I be a family of finite real numbers; then the following statements are equivalent: ↩ 2 0BTG
Proposition 1 If the families (x_λ)_λ ∈ L and (y_μ)_μ ∈ M of finite real numbers are summable in R, then so is the family (x_λ y_μ)_(λ, μ) ∈ L × M and we have ↩ 2 0BTH
Theorem 4 The family (1 + uᵢ) is multipliable in R^* if and only if the family (uᵢ) is summable in R. ↩ 7 0BTI
Lemma (i) If (aᵢ)_1 ≤ i ≤ p is a finite sequence of numbers > 0, then ∏ᵢ₌₁ᵖ (1 + aᵢ) ≥ 1 + ∑ᵢ₌₁ᵖ aᵢ. (ii) If also aᵢ < 1 for each i, then ∏ᵢ₌₁ᵖ (1 - aᵢ) ≥ 1 - ∑ᵢ₌₁ᵖ aᵢ. 0BTJ
Proposition 2 Every family (xᵢ) of real numbers ≥ 0 is summable in overlineR. ↩ 1 0BTK
Proposition 3 Every family (1 + uᵢ) [resp. (1 - uᵢ)] of numbers ≥ 1 (resp. ≥ 0 and ≤ 1) is multipliable in overlineR. 0BTL
Corollary The product ∏ (1 + uᵢ) [resp. ∏ (1 - uᵢ)] of numbers ≥ 1 (resp. > 0 and ≤ 1) is equal to + ∞ (resp. 0) if and only if ∑ uᵢ = + ∞. 0BTM
Remark The theorem of associativity (Chapter III, § 5, no. 3, Theorem 2) remains valid when G is replaced by overlineR and the xᵢ are assumed to be ≥ 0. This is clear if ∑_i ∈ I xᵢ is fin… 0BTN
Definition 1 A series of finite real numbers is said to be absolutely convergent if the series of absolute values of its terms is convergent. 0BTO
Proposition 4 A series of finite real numbers is commutatively convergent if and only if it is absolutely convergent. 0BTP
Example Alternating series. A series defined by a sequence (uₙ) of finite real numbers is called alternating if uₙ = (-1)ⁿ vₙ, where vₙ ≥ 0 for all n. Let us show that a sufficient conditi… 0BTQ
Definition 2 An infinite product whose general factor is 1 + uₙ is said to be absolutely convergent if the product whose general factor is 1 + |uₙ| is convergent. 0BTR
Proposition 5 An infinite product of finite real numbers is commutatively convergent if and only if it is absolutely convergent. 0BTS
Example If we take u₂ₙ₋₁ = -1/n and u₂ₙ = 1/n for n ≥ 2, the product (1 + uₙ) is not absolutely convergent, since the series (uₙ) is not absolutely convergent; but since 0BTT

§ 8. Usual expansions of real numbers; the power of R

Definition 1 Given a number ε > 0, a real number r is said to be an approximation to within ε to a real number x, if |x - r| ≤ ε; r is said to be an approximation by defect if r ≤ x, by excess… 0BUG
Theorem 1 (Cantor) The set of real numbers is equipotent to the set of all subsets of a countably infinite set. ↩ 1 0BUH
Corollary The set of real numbers has a cardinal strictly greater than the cardinal of a countable set. 0BUI

V. One-parameter groups

§ 1. Subgroups and quotient groups of R

Proposition 1 Every closed subgroup of the additive group R, other than R and 0, is a discrete group of the form a.Z, where a > 0 (in other words, it consists of the integer multiples of a). ↩ 2 0BV0
Proposition 2 The Hausdorff quotient groups of R, other than 0, are the groups R/aZ (a geqslant 0). If a and b are > 0, the automorphism x → b/a x of R transforms aZ into bZ, and therefore (Chap… 0BV1
Proposition 3 Every Hausdorff quotient group of R, other than R and 0, is isomorphic to the group R/Z. ↩ 1 0BV2
Definition 1 The topological group R/aZ (a > 0) is called the additive group of real numbers modulo a. The topological group R/Z is denoted by T. As a topological space, T is called the one-dim… 0BV3
Remark 1 The relation x ≡ y pmodaZ is usually written x ≡ y pmoda, or simply x ≡ y (a), and is read "x and y are congruent modulo a"; it means that x - y is an integer multiple of a. When a… 0BV4
Remark 2 \ As we shall see in Chapter VI, § 2, no. 4, the topological space T is homeomorphic to the circle x² + y² = 1 in the real number plane R²; the product space T² is homeomorphic to… 0BV5
Proposition 4 The torus T is homoeomorphic to the quotient space of any closed interval of R of the form [a, a+1] obtained by identifying the end-points of this interval; it is compact, connecte… ↩ 1 0BV6
Remark Note that the canonical mapping φ of R onto T = R/Z, restricted to the half-open interval [a, a+1[, is a continuous bijective mapping of this interval onto T; the inverse mapping i… 0BV7
Proposition 5 Every continuous homomorphism f of the topological group R into itself is of the form x → ax, where a ∈ R; it is an automorphism of R if a ≠ 0. ↩ 1 0BV8
Corollary Let G be a topological group isomorphic to R. For each a ∈ G there is exactly one continuous homomorphism fₐ of R into G such that fₐ(1) = a, and this homomorphism is an isomorphis… ↩ 1 0BV9
Proposition 6 Let I be an interval of R which contains o and at least one other point; let f be a continuous mapping of I into a topological group G (written multiplicatively), such that f(x+y)… ↩ 3 0BVA
Corollary Let f be a local isomorphism of R with a topological group G. Then there is a unique strict morphism of R onto an open subgroup of G which coincides with f at all points of some ne… ↩ 1 0BVB
Proposition 7 Every connected group G which is locally isomorphic to R is isomorphic to either R or T. ↩ 1 0BVC

§ 2. Measurement of magnitudes

Proposition 1 Let E be a linearly ordered set with a smallest element ω; let I be a subset of E such that ω ∈ I and such that the relations x ∈ I, y ≤ x imply y ∈ I; let (x, y) → xy be a law of… ↩ 3 0BVG
Remark 1 The relations x ∈ I, y ∈ I, xy ∈ I, yx ∈ I imply f(xy) = f(x) + f(y) = f(yx), and hence yx = xy since f is strictly increasing; in other words, the law induced by the law of compos… 0BVH
Remark 2 Every mapping g of I into R₊ which satisfies the same conditions as f is of the form x → λ f(x) where λ > 0. For if λ = g(a) > 0, the relations zᵖ ≤ x ≤ zᵖ⁺¹, z^q ≤ a ≤ z^q+1 imply… 0BVI
Proposition 2 If a linearly ordered set E and a subset I of E satisfy the axioms (GRI), (GR{II}), (GR{IIIa}) and (GR{IVa}), there exists a strictly increasing mapping f of I onto an interval of… ↩ 2 0BVJ
Remark When I = E, the image f(I) = f(E) is the whole of R₊, for if b > ω, then bⁿ is defined for all n, and therefore n · f(b) belongs to f(E) for all n; this implies that f(E) is not bo… 0BVK

§ 3. Topological characterization of the groups R and T

Theorem 1 A topological group G in which there exists a neighbourhood of the identity element homeomorphic to an open interval of R is locally isomorphic to R. ↩ 1 0BVN
Theorem 2 A connected group G in which there exists a neighbourhood of the identity element homeomorphic to an open interval of R is isomorphic to either R or T. ↩ 3 0BVO
Remark 1 To decide whether a group G which satisfies the conditions of Theorem 2 is isomorphic to T or to R, it is enough to see whether G is compact or not. 0BVP
Remark 2 Theorem 2 shows in particular that every topological group which is homeomorphic to the group R is necessarily isomorphic to R. 0BVQ
Remark 3 The preceding topological characterization of the groups R and T involves the topological space R as an auxiliary set. It is possible to characterize the topological group structur… 0BVR

§ 4. Exponentials and logarithms

Theorem 1 The multiplicative group R₊^* of real numbers > 0 is a topological group isomorphic to the additive group R of real numbers. ↩ 1 0BW0

VI. Real number spaces and projective spaces

§ 1. Real number space R^n

Definition 1 The topological product of n spaces identical with the real line is called real number space of n dimensions, and is denoted by Rⁿ. 0BW3
Remark The space R⁰ consists of a single point. 0BW4
Definition 2 Any subset of Rⁿ which is the product of n open (resp. closed) intervals of R is called an open (resp. closed) box in Rⁿ. [For n = 2 it is called an open (resp. closed) rectangle.] 0BW5
Proposition 1 A subset A of Rⁿ is relatively compact if and only if it is bounded. 0BW6
Corollary The space Rⁿ is locally compact, but is not compact if n ≥ 1. 0BW7
Proposition 2 Every p-dimensional linear affine variety in Rⁿ is a closed subset of Rⁿ, homeomorphic to Rᵖ. ↩ 1 0BW8
Proposition 3 Let f(x) = f(x₁, x₂, …, xₙ) be a polynomial with real coefficients, not identically zero, defined on Rⁿ. Then the complement of the set overlinef(0) is dense in Rⁿ. ↩ 1 0BW9
Corollary The complement of an affine linear variety of dimension p < n is dense in Rⁿ. ↩ 1 0BWA
Proposition 4 In Rⁿ (n ≥ 1) the complement of every hyperplane has two connected components. 0BWB
Proposition 5 Let E, F, G be three finite-dimensional vector spaces over the field R. Then every bilinear mapping () f of E × F into G is continuous. 0BWC
Proposition 6 In the ring Mₙ(R), the group GLₙ(R) of non-singular matrices is a dense open subset, and the topology induced on this set is compatible with its group structure. ↩ 2 0BWD

§ 2. Euclidean distance, balls and spheres

Definition 1 Given a point x₀ ∈ Rⁿ and a real number r > 0, the open (resp. closed) Euclidean ball of n dimensions with centre x₀ and radius r is the set of all points x ∈ Rⁿ such that d(x₀, x)… 0BWR
Proposition 1 Every open (resp. closed) ball of Rⁿ is an open (resp. compact) set. The closure of an open ball is the closed ball with the same centre and the same radius; the interior of a clos… 0BWS
Corollary Every Euclidean sphere is a compact set and is the frontier of the open and closed balls with the same centre and the same radius. 0BWT
Proposition 2 Every n-dimensional open ball is homeomorphic to Rⁿ. ↩ 1 0BWU
Proposition 3 The space Rₙ^* is homeomorphic to the product of Sₙ₋₁ and the space R₊^* of real numbers > 0. ↩ 4 0BWV
Corollary 1 The sphere Sₙ₋₁ is homeomorphic to the quotient of Rₙ^* by the equivalence relation whose classes are the open rays with origin o. 0BWW
Corollary 2 The space Rₙ^* is homeomorphic to R × Sₙ₋₁. 0BWX
Remark These propositions are not peculiar to Euclidean balls, but can be extended to a whole category of compact neighbourhoods of o in Rⁿ (see Exercise 12). 0BWY
Proposition 4 If n > 1, the Euclidean sphere Sₙ₋₁ is homeomorphic to the space Rⁿ⁻¹ made compact by adjoining a "point at infinity"(Chapter I, § 9, no. 8, Theorem 4). ↩ 4 0BWZ
Corollary 1 The sphere Sₙ is homeomorphic to the quotient space of the ball Bₙ obtained by identifying all the points of the sphere Sₙ₋₁. 0BX0
Corollary 2 The circle S₁ is homeomorphic to the torus T. ↩ 1 0BX1
Proposition 5 If n > 1, the Euclidean sphere Sₙ₋₁ is connected and locally connected, and every point of it has an open neighbourhood homeomorphic to Rⁿ⁻¹. ↩ 2 0BX2

§ 3. Real projective spaces

Lemma 1 If every open (resp. closed) set in A which is saturated with respect to R_A is the trace on A of an open (resp. closed) set in E which is saturated with respect to R, then the quo… ↩ 4 0BXG
Lemma 2 If there is a continuous mapping g of E onto A such that, for all x ∈ E, g(x) belongs to the equivalence class of x, then the quotient space A/RA is homeomorphic to E/R. ↩ 4 0BXH
Definition 1 The projective space Pn(R) endowed with the quotient topology of the topology of R{n+1}^ by the equivalence relation Δn(R) is called real projective space of n dimensions. 0BXI
Proposition 1 The projective space Pₙ is Hausdorff. ↩ 1 0BXJ
Proposition 2 The projective space Pₙ is compact and connected, and is homeomorphic to the quotient of the sphere Sₙ by the equivalence relation induced on the sphere by Δₙ. ↩ 2 0BXK
Proposition 3 If n ≥ 0, the projective space Pₙ is homeomorphic to the quotient space of the ball Bₙ obtained by identifying each point of Sₙ₋₁ with its diametrically opposite point. ↩ 1 0BXL
Proposition 4 Every projective linear variety of p dimensions in a projective space Pₙ is closed in Pₙ and homeomorphic to Pₚ; if p < n its complement is dense in Pₙ. 0BXM
Proposition 5 In a projective space Pₙ (n ≥ 0), the complement of a projective hyperplane H is homeomorphic to Rⁿ. ↩ 2 0BXN
Corollary Every point of Pₙ has an open neighbourhood homeomorphic to Rⁿ. 0BXO
Definition 2 The space P_n,\ p which is the quotient of the topological space L_n+1,\ p+1 by the equivalence relation Δ_n,\ p is called the space of projective linear varieties of p ≥ 0 dimensi… ↩ 1 0BXP
Proposition 6 The space P_n,\ p is Hausdorff. ↩ 2 0BXQ
Proposition 7 The space P_n,p is compact. ↩ 1 0BXR
Proposition 8 P_n,p is connected and locally connected, and every point has an open neighbourhood homeomorphic to R^(p+1)(n-p). ↩ 2 0BXS
Proposition 9 The Grassmannian G_n,p is homeomorphic to the space P_n,p. ↩ 2 0BXT

VII. The additive groups $\mathbf{R}^n$

§ 1. Subgroups and quotient groups of $\mathbf{R}^n$

Proposition 1 Let G be a discrete subgroup of Rⁿ of rank p, let (aᵢ)_1 ≤ i ≤ p be a free system of p points of G, and let P be the closed parallelo-tope with centre o and basis vectors aᵢ (Chapt… ↩ 3 0BY4
Corollary Let (aᵢ)_1 ≤ i ≤ p be a free system of p points of Rⁿ and let b = ∑ᵢ₌₁ᵖ tᵢ aᵢ be a linear combination of the aᵢ, with real coefficients. Then the subgroup G of Rⁿ generated by the… 0BY5
Proposition 2 (Kronecker) Let θ₁, θ₂, …, θₙ be n real numbers. In order that, for each ε > 0, there exist an integer q and n integers pᵢ (1 ≤ i ≤ n) such that |qθᵢ - pᵢ| ≤ ε (1 ≤ i ≤ n), where the left-hand… ↩ 1 0BY6
Theorem 1 Every discrete subgroup G of Rⁿ, of rank equal to p, is generated by a free system of p points. ↩ 4 0BY7
Proposition 3 Every non-discrete closed subgroup of Rⁿ contains a line passing through o. ↩ 1 0BY8
Theorem 2 Let G be a closed subgroup of Rⁿ, of rank r (0 ≤ r ≤ n). Then there is a largest vector subspace V contained in G; for every vector subspace W complementary to V, W ∩ G is discrete… ↩ 5 0BY9
Corollary 1 There exists a basis (aᵢ)_1 ≤ i ≤ n of Rⁿ, such that aᵢ ∈ G (1 ≤ i ≤ r), aᵢ ∈ V (1 ≤ i ≤ p) and such that G is the set of points ∑ᵢ₌₁ᵖ tᵢ aᵢ + ∑_j=p+1ʳ nⱼ aⱼ, where the tᵢ take all… ↩ 2 0BYA
Corollary 2 There exists an automorphism of Rⁿ which maps G onto the group G', isomorphic to Rᵖ × Zʳ⁻ᵖ, which is the direct sum of the vector subspace generated by e₁, e₂, …, eₚ and the (discr… 0BYB
Proposition 4 The subgroup G^* associated with a subgroup G of Rⁿ is closed, and we have (overlineG)^* = G^*. 0BYC
Proposition 5 For every closed subgroup G of Rⁿ, r(G^*) = n - d(G) and d(G^*) = n - r(G). Let us apply the same reasoning to G^*. Observing that the basis dual to (a'ᵢ) is (aᵢ), we see that: ↩ 1 0BYD
Proposition 6 For every subgroup G of Rⁿ, we have (G^*)^* = overlineG. ↩ 2 0BYE
Corollary A point x lies in the closure of a subgroup G of Rⁿ if and only if (u|x) is an integer for all u ∈ Rⁿ such that (u|y) is an integer for all y ∈ G. 0BYF
Proposition 7 (Kronecker) Let aᵢ = (aᵢⱼ) (1 ≤ i ≤ m,\ 1 ≤ j ≤ n) be m points of Rⁿ and let b = (bⱼ) (1 ≤ j ≤ n) be a point of Rⁿ. In order that, for each ε > 0, there exist m integers gᵢ (1 ≤ i ≤ m) and n i… 0BYG
Corollary 1 In order that, for each x = (xⱼ) (1 ≤ j ≤ n) and each ε > 0, there exist m integers qᵢ (1 ≤ i ≤ m) and n integers pⱼ (1 ≤ j ≤ n) such that ↩ 1 0BYH
Corollary 2 Let θ₁, θ₂, …, θₙ be n real numbers. In order that, given any n real numbers x₁, x₂, …, xₙ and a real number ε > 0, there should exist an integer q and n integers pⱼ such that ↩ 3 0BYI
Proposition 8 If G₁, G₂ are any closed subgroups of Rⁿ, we have 0BYJ
Remark Let G₁, G₂ be two lattices in Rⁿ (no. 1) such that G₂ ⊂ G₁: then (Proposition 5) G₁^* and G₂^* are lattices in Rⁿ such that G₁^* ⊂ G₂^*. We have seen in no. 1 that there is an inte… 0BYK
Proposition 9 Every Hausdorff quotient group of Rⁿ is isomorphic to a product group Rʰ × Tᵏ (0 ≤ h + k ≤ n). ↩ 1 0BYL
Proposition 10 The topological group Tⁿ is locally isomorphic to Rⁿ. 0BYM
Corollary Every closed subgroup of T, other than T itself, is a finite cyclic group. Every Hausdorff quotient group of T, other than 0, is isomorphic to T. 0BYN

§ 2. Continuous homomorphisms of $\mathbf{R}^n$ and its quotient groups

Proposition 1 Every continuous homomorphism f of the additive group Rᵐ into the additive group Rⁿ is a linear mapping of Rᵐ into Rⁿ. ↩ 1 0BZ3
Proposition 2 Let A be a parallelopte in Rⁿ which contains o; and let f be a continuous mapping of A into a topological group G (written multiplicatively) such that f(x + y) = f(x)f(y) for each… ↩ 3 0BZ4
Corollary 1 Let V be a neighbourhood of o in Rⁿ and let f be a continuous mapping of V into a topological group G such that ↩ 1 0BZ5
Corollary 2 Let f be a local isomorphism of Rⁿ with a topological group G. Then there exists a unique strict morphism of Rⁿ onto an open subgroup of G which agrees with f at all the points of… ↩ 1 0BZ6
Theorem 1 Every connected group G which is locally isomorphic to Rⁿ is isomorphic to a group Rᵖ × Tⁿ⁻ᵖ (0 leqslant p leqslant n). ↩ 1 0BZ7
Proposition 3 Every continuous homomorphism of Rᵐ into Tⁿ is of the form x → φ(u(x)), where φ is the canonical homomorphism of Rⁿ onto Tⁿ (identified with Rⁿ/Zⁿ) and u is a linear mapping of Rᵐ… ↩ 2 0BZ8
Remark The same argument shows, more generally, that if φ is a strict morphism of Rⁿ into a group G, whose restriction to a suitable neighbourhood of o is a local isomorphism of Rⁿ with G… 0BZ9
Proposition 4 If φ is the canonical homomorphism of R onto T, every continuous homomorphism of R into T is of the form x ↦ φ(ax) where a ∈ R; and it is a strict morphism of R onto T if a ≠ 0. ↩ 1 0BZA
Proposition 5 Every isomorphism of the topological group Tⁿ onto one of its subgroups is an automorphism of Tⁿ which is obtained by passing to the quotient from a linear mapping u of Rⁿ onto its… ↩ 3 0BZB
Proposition 6 The only isomorphisms of the topological group T onto one of its subgroups are the identity mapping and the symmetry x → -x. ↩ 3 0BZC

§ 3. Infinite sums in the groups $\mathbf{R}^n$

Proposition 1 A family (xᵢ)_i ∈ I of points xᵢ = (x_i,k)_1 ≤ k ≤ n of Rⁿ is summable if and only if each of the n families (x_i,k)_i ∈ I of real numbers is summable in R. ↩ 2 0BZK
Theorem 1 A family (xᵢ)_i ∈ I of points of Rⁿ is summable if and only if the family (||xᵢ||) of Euclidean norms of the xᵢ is summable in R. ↩ 2 0BZL
Proposition 2 If (xᵢ)_i ∈ I is any finite family of points in Rⁿ, then ↩ 3 0BZM
Corollary A family (xᵢ) of points of Rⁿ is summable if and only if the set of finite partial sums of the family is bounded in Rⁿ. 0BZN
Proposition 3 Let (x_λ)_λ ∈ L be a summable family of points of Rᵐ, (y_μ)_μ ∈ M a summable family of points of Rⁿ, and let f be a bilinear mapping of Rᵐ × Rⁿ into Rᵖ. Then the family (f(x_λ, y_μ… 0BZO
Corollary 1 If (a_λ)_λ ∈ L is a summable family of real numbers and if (x_μ)_μ ∈ M is a summable family of points of Rⁿ, then the family (a_λ x_μ)_(λ, μ) ∈ L × M is summable and we have 0BZP
Corollary 2 If (x_λ)_λ ∈ L and (y_μ)_μ ∈ M are two summable families of points of Rⁿ, then the family (x_λ | y_μ) (cf. Chapter VI, § 2, no. 2) is summable in R, and we have 0BZQ
Definition 1 A series of points of Rⁿ is said to be absolutely convergent if the series of Euclidean norms of its terms is convergent. 0BZR
Proposition 4 A series of points of Rⁿ is commutatively convergent if and only if it is absolutely convergent. 0BZS

VIII. Complex numbers

§ 1. Complex numbers, quaternions

Definition 1 The field R[X]/(X² + 1) is called the field of complex numbers and is denoted by C. The canonical image of X in C is denoted by i, so that C is obtained from the field R by algebra… ↩ 1 0BZV
Theorem 1 (d'Alembert-Gauss). The field C of complex numbers is algebraically closed. ↩ 2 0BZW
Remark 1 Theorem 1 can be proved without invoking the theory of ordered fields by using properties of the topology of the field C, which will be defined below (no. 2); see § 2, Exercise 2 a… 0BZX
Remark 2 Since C is of degree 2 over R, it follows that C is, up to isomorphism, the only algebraic extension of R other than R itself, and that there is no field contained in C which conta… 0BZY
Proposition 1 The topological group C^* is isomorphic to the product of the topological groups R^* and U. ↩ 3 0BZZ
Proposition 2 The multiplicative group H^* of non-zero quaternions is a topological group isomorphic to the product of its subgroups R₊^* and S₃. 0C00
Remark 1 With the use of the relations |x + y| = |x| + |y| and |xy| = |x|·|y| it can be proved directly, as with the field of complex numbers in no. 2, that the topology of R⁴ is compatible… 0C01
Remark 2 From what has been proved it follows that the spheres S₁ and Sₙ can carry a group structure compatible with their topology. We shall see later that, for each integer n other than 1… 0C02
Remark 3 Every point of the group S₂ has a neighbourhood homeomorphic to R³ (Chapter VI, § 2, no. 4, Proposition 5) but S₄ is not locally isomorphic to R³; for if it were it would be abelia… 0C03

§ 2. Angular measure, trigonometric functions

Theorem 1 The (multiplicative, topological group U of complex numbers of absolute value 1 is isomorphic to the (additive, topological group T of real numbers mod 1. ↩ 2 0C09
Corollary The multiplicative group C^* of non-zero complex numbers is isomorphic to the group R × T (cf. § 1, no. 3, Proposition 1). 0C0A
Remark The isomorphism of the groups C^* and R × T implies the existence of roots of every "binomial equation" zⁿ = a in the field C. Using this fact and the local compatibility of C, we… 0C0B
Remark The mapping x + iy → eˣe(y) is a strict morphism of the additive group C onto the multiplicative group C^*, and its restriction to a suitable neighbourhood of 0 is a local isomorph… 0C0C
Remark We can define a homomorphism of C^* onto 𝔄₀ by mapping each complex number z ≠ 0 to the cross which the line through o and z makes with the real axis. Clearly this homomorphism is… 0C0D

§ 3. Infinite sums and products of complex numbers

Proposition 1 If (u_λ)_λ ∈ L and (v_μ)_μ ∈ M are two summable families of complex numbers, then the family (u_λ v_μ)_(λ, μ) ∈ L × M is summable and we have 0C0I
Proposition 2 A family (zᵢ) of complex numbers zᵢ = rᵢ (cos θᵢ + i sin θᵢ) is multipliable if and only if the family (rᵢ) of absolute values of the zᵢ is multipliable in R₊^* and the family (θᵢ)… 0C0J
Theorem 1 The family (1 + uᵢ)_i ∈ I is multipliable in C^* if and only if the family (|uᵢ|) is summable in R. ↩ 1 0C0K
Lemma 1 For each finite subset J of I, let φ(J) = sup_L ⊂ J (p_L - 1). Then for each subset L of J we have 0C0L
Lemma 2 If J is a finite subset of I such that φ(J) < 1/4, then 0C0M
Definition 1 An infinite product of complex numbers, with general factor 1 + uₙ, is said to be absolutely convergent if the product with general factor 1 + |uₙ| is convergent (or, equivalently… 0C0N
Proposition 3 An infinite product of complex numbers is commutatively convergent if and only if it is absolutely convergent. ↩ 1 0C0O
Remark 1 The product with general factor |1 + uₙ| can be convergent, and indeed absolutely convergent in Rᵢ^*, without the convergence of the product with general factor 1 + |uₙ| (see Exerc… 0C0P
Remark 2 As already remarked for products of factors > 0, the convergence of the series with general term uₙ is neither necessary nor sufficient for the convergence of the product with gene… 0C0Q

§ 4. Complex number spaces and projective spaces

Definition 1 The projective space Pₙ(C), endowed with the topology which is the quotient of that of Cₙ₊₁^* by the equivalence relation Δₙ(C), is called complex projective space of n dimensions. 0C0W
Definition 2 The quotient space P_n,p(C) of the topological space L_n+1,\ p+1(C) by the equivalence relation Δ_n,p(C) is called the space of projective linear varieties of p geqslant 0 dimensio… ↩ 1 0C0X
Remark Most of the properties common to the real and complex number spaces (resp. projective spaces) are also valid for number spaces (resp. projective spaces) defined in the same way ove… 0C0Y

IX. Use of real numbers in general topology

§ 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces

Example 1 On real number space Rⁿ, Euclidean distance (Chapter VI, § 2, no. 1) is a pseudometric. 0C17
Example 2 If X is any set, the function f defined on X × X by the conditions f(x, x) = 0 for all x ∈ X, f(x, y) = +∞ if x ≠ y is a pseudometric on X. ↩ 1 0C18
Example 3 If X is any set and if g is any finite real-valued function defined on X, then the function f defined on X × X by f(x, y) = |g(x) - g(y)| is a pseudometric on X. ↩ 1 0C19
Example 4 \ Let X be the set of all continuous mappings of the interval [0, 1] of R into R. If for each pair of elements x, y of X we put f(x, y) = ∫₀¹ |x(t) - y(t)| dt, then f is a pseudome… 0C1A
Definition 2 Given a pseudometric f on a set X, the uniformity defined by f is the uniformity on X which has as a fundamental system of entourages the family of sets overlinef⁻¹([0, a]), where… ↩ 2 0C1B
Remark 1 If (aₙ) is any sequence of numbers > 0 and tending to 0, the U_aₙ form a fundamental system of entourages of the uniformity defined by f. 0C1C
Remark 2 The definition of a uniformity by a pseudometric f consists in taking as a fundamental system of entourages the inverse image under f of the neighbourhood filter of 0 in the subspa… 0C1D
Definition 3 If (fᵢ)_i ∈ I is a family of pseudometrics on a set X, then the least upper bound of the set of uniformities defined on X by the pseudometrics fᵢ is called the uniformity defined b… 0C1E
Proposition 1 Let X be a Hausdorff uniform space whose uniformity U is defined by a family of finite pseudometrics (fᵢ), and let hatX be the completion of X. Then the functions fᵢ can be extende… ↩ 2 0C1F
Theorem 1 Given a uniformity U on a set X, there is a family of pseudometrics on X such that the uniformity defined by this family is identical with U. ↩ 2 0C1G
Proposition 2 If a uniformity U on X has a countable fundamental system of entourages, then there is a pseudometric f on X such that U is identical with the uniformity defined by f. ↩ 2 0C1H
Remark A uniformity U on X is defined by the family Φ of all pseudometrics on X which are uniformly continuous on X × X. For clearly the uniformity defined by the family Φ is coarser than… 0C1I
Theorem 2 A topological space X is uniformizable if and only if it satisfies the following axiom: ↩ 4 0C1J
Definition 4 A topological space is said to be completely regular if it is uniformizable and Hausdorff. 0C1K
Remark Axiom (O{IV}) implies (O{III}) (cf. Chapter I, § 8, no. 4), for if V is a neighbourhood of x0 and if f is a continuous real-valued function on X with values in [0, 1], such that f(… 0C1L
Proposition 3 A topological space X is completely regular if and only if it is homeomorphic to a subspace of a compact space. ↩ 1 0C1M
Proposition 4 If a topological space X is completely regular, it is homeomorphic to a subspace of a cube. ↩ 1 0C1N
Proposition 5 In order that every lower semi-continuous real-valued function f (finite or not) on a topological space X should be the upper envelope of the continuous real-valued functions on X… ↩ 1 0C1O
Remark In the above proof we see that the function g does not take the value +1. By transfer of structure it follows that every lower semi-continuous real-valued function f on the uniform… 0C1P

§ 2. Metric spaces and metrizable spaces

Definition 1 A metric on a set X is a finite pseudometric d on X such that the relation d(x, y) = 0 implies x = y. A metric space is a set X endowed with the structure defined by a given metric… ↩ 1 0C2C
Example 1 The Euclidean distance d(x, y) (Chapter VI, § 2, no. 1) is a metric on real n-dimensional number space Rⁿ; so are the functions 0C2D
Example 2 On any set X the pseudometric d, defined by the relations d(x, x) = 0 and d(x, y) = 1 if x ≠ y, is a metric. The uniformity it defines on X is the discrete uniformity. 0C2E
Proposition 1 Let X be a metric space and let d be its metric. If hatX is the completion of X (with respect to the uniformity defined by d), the function d can be extended by continuity to hatX… ↩ 3 0C2F
Remark If d(x, A) = a, it can happen that there is no point of A whose distance from x is equal to a. However, this situation can never arise if A is compact, for then Weierstrass's theor… 0C2G
Proposition 2 The statements d(x, A) = 0 and x ∈ overlineA are equivalent. ↩ 2 0C2H
Proposition 3 The function d(x, A) is uniformly continuous on X. ↩ 3 0C2I
Remark We can have d(A, B) = 0 for two subsets A, B of X such that overlineA ∩ overlineB = ∅, provided that neither subset consists of a single point. For example, on the real line R, the… 0C2J
Proposition 4 The oscillation ω(x; f) of an arbitrary function f, defined on a subset X of a topological space Y and taking its values in a metric space X', is upper semi-continuous on overlineX… 0C2K
Proposition 5 Let f be a function defined on a subset X of a topological space Y, taking its values in a complete metric space X'. Then f has a limit relative to X at a point x ∈ overlineX if an… 0C2L
Definition 2 A metric on a set X is said to be compatible with a uniformity U on X if the uniformity defined by the metric coincides with U. A uniformity on a set X is said to be metrizable if… 0C2M
Theorem 1 A uniformity is metrizable if and only if it is Hausdorff and the filter of entourages of the uniformity has a countable base. ↩ 8 0C2N
Corollary 1 A Hausdorff uniformity defined by a countable family of pseudometrics is metrizable. 0C2O
Corollary 2 Every countable product of metrizable uniform spaces is metrizable. ↩ 5 0C2P
Definition 3 A metric on a set X is said to be compatible with a topology T on X if the topology defined by this metric coincides with T. A topological space is said to be metrizable if there e… 0C2Q
Proposition 6 Every point of a metrizable space has a countable fundamental system of neighbourhoods. ↩ 1 0C2R
Proposition 7 In a metrizable space, every closed set is the intersection of a countable family of open sets, and every open set is the union of a countable family of closed sets. ↩ 2 0C2S
Remark 1 These necessary conditions are not sufficient (cf. Exercise 13). 0C2T
Remark 2 There are spaces in which every point has a countable fundamental system of neighbourhoods but in which there exist closed sets which are not countable intersections of open sets (… 0C2U
Proposition 8 In a metrizable space X, a point x lies in the closure of a non-empty subset A of X if and only if there is a sequence of points of A which converges to x. ↩ 1 0C2V
Proposition 9 A metric space X is complete if and only if every Cauchy sequence in X is convergent. ↩ 3 0C2W
Proposition 10 Let X be a metrizable space and let f be a mapping of X into a topological space X'. Then f is continuous at a point x ∈ X if and only if, whenever (xₙ) is a sequence of points of… ↩ 1 0C2X
Proposition 11 Let X be a metrizable space and let f be a lower semi-continuous function on X which takes its values in a closed interval [a, b] of overlineR. Then f is the upper envelope of an i… ↩ 1 0C2Y
Definition 4 A metrizable space is said to be of countable type (or separable) if its topology has a countable base. 0C2Z
Proposition 12 If X is a metrizable topological space, the following statements are equivalent: ↩ 3 0C30
Proposition 13 Let X be a topological space which has a countable base (Uₙ); then for each open covering (Vᵢ)_i ∈ I of X there exists a countable subset J of I such that (Vᵢ)_i ∈ J is a covering… 0C31
Proposition 14 A metric space X is precompact if and only if, for each ε > 0, there is a finite covering of X by sets of diameter ≤ ε. ↩ 2 0C32
Proposition 15 A metrizable topological space X is compact if and only if every infinite sequence of points of X has a cluster point in X. ↩ 2 0C33
Corollary A subset A of a metrizable topological space X is relatively compact if and only if every infinite sequence of points of A has a cluster point in X. 0C34
Proposition 16 A compact space X is metrizable if and only if its topology has a countable base. ↩ 4 0C35
Corollary Let X be a locally compact space and let X' be the compact space obtained by adjoining a point at infinity ω to X (Chapter I, § 9, no. 8). Then the following statements are equival… ↩ 1 0C36
Proposition 17 Every Hausdorff quotient space of a compact metrizable space is compact and metrizable. 0C37
Proposition 18 Let X be a complete metric space, let R be an open equivalence relation on X such that X/R is Hausdorff, and let φ : X → X/R be the canonical mapping. Then if K is any compact subs… 0C38

§ 3. Metrizable groups, valued fields, normed spaces and algebras

Proposition 1 The left and right uniformities of a topological group G are metrizable if and only if G is Hausdorff and the identity element e of G has a countable fundamental system of neighbou… ↩ 1 0C3Y
Definition 1 A metric d on a group G (written multiplicatively) is said to be left-invariant (resp. right-invariant) if we have 0C3Z
Proposition 2 The left (resp. right) uniformity of a metrizable group G can be defined by a left-invariant (resp. right-invariant) metric on G. 0C40
Proposition 3 Let G be an abelian group, written additively, and let x → |x| be a mapping G → R which satisfies conditions a), b) and c) above. Then the function d(x, y) = |x - y| is an invarian… ↩ 2 0C41
Proposition 4 If G is a metrizable group, then every Hausdorff quotient group G/H of G is metrizable. If G is also complete, then G/H is complete (). ↩ 4 0C42
Corollary 1 Let G be a complete metrizable group, let G0 be a dense subgroup of G and let H0 be a closed normal subgroup of G0. If H is the closure of H0 in G, the quotient group G0/H0 has a c… ↩ 1 0C43
Corollary 2 Let G, G' be two metrizable abelian topological groups; let u : G → G' be a strict morphism with kernel N and image P. Then hatu : hatG → hatG' is a strict morphism with kernel hat… ↩ 1 0C44
Corollary 3 Let G, G', G'' be three metrizable abelian topological groups and let u : G → G' and v : G' → G'' be two strict morphisms such that the sequence G xrightarrowu G' xrightarrowv G''… 0C45
Remark 1 Let G be a non-Hausdorff topological group such that the Hausdorff group associated with G is metrizable; equivalently, such that the identity element of G has a countable fundamen… 0C46
Remark 2 Let d be a left-invariant metric which defines the topology of a metrizable group G, and let H be a closed normal subgroup of G. If dotx and doty are any two points of G/H, conside… 0C47
Definition 2 An absolute value on a division ring K is a mapping x → |x| of K into R₊ which satisfies the following conditions: ↩ 1 0C48
Definition 3 A valued division ring is a division ring K endowed with the structure defined by a given absolute value on K. ↩ 1 0C49
Example 1 Let K be an arbitrary division ring. For each x ∈ K, put |x| = 1 if x ≠ 0, and |0| = 0. The mapping x → |x| so defined is an absolute value on K, called the improper absolute value… 0C4A
Example 2 The absolute value of a real number (Chapter IV, § 1, no. 6) satisfies axioms (VMI), (VM{II}) and (VM{III}), and the topology it defines on the field R is the topology of the real… 0C4B
Example 3 On a division ring K, a real valuation is a function v defined on K^* with values in R which satisfies the following conditions: a) if x ∈ K^* and y ∈ K^*, then v(xy) = v(x) + v(y)… 0C4C
Remark If x is a root of unity in a valued division ring, then |x| = 1, for xⁿ = 1 implies |x|ⁿ = 1 and hence |x| = 1. In particular, the only absolute value on a finite field is the impr… 0C4D
Definition 4 Two absolute values on a division ring K are said to be equivalent if they define the same topology on K. 0C4E
Proposition 5 Two absolute values |x|₁, |x|₂ on a division ring K, neither of which is the improper absolute value, are equivalent if and only if the relation |x|₁ < 1 implies |x|₂ < 1. There ex… ↩ 6 0C4F
Remark The proof of Proposition 5 shows that, if the topology defined by |x|₂ is coarser than that defined by |x|₁, and if |x|₁ is not improper, then |x|₁ and |x|₂ are equivalent, for the… ↩ 1 0C4G
Proposition 6 The completion hatK of a division ring K endowed with an absolute value |x| is a division ring, and the function |x| can be extended by continuity to an absolute value on hatK, whi… ↩ 1 0C4H
Definition 5 If E is a (left) vector space over a non-discrete valued division ring K, a norm on E is a mapping x → p(x) of E into R₊ which satisfies the following axioms: 0C4I
Proposition 7 Two norms p, q on a vector space E are equivalent if and only if there exist two numbers a > 0, b > 0 such that ↩ 3 0C4J
Example In the space Rⁿ, the three norms sqrt∑ᵢ₌₁ⁿ xᵢ², sup_1 ≤ i ≤ n |xᵢ| and ∑ᵢ₌₁ⁿ |xᵢ| are equivalent, because we have (2) sup_1 ≤ i ≤ n |xᵢ| ≤ sqrt∑ᵢ₌₁ⁿ xᵢ² ≤ ∑ᵢ₌₁ⁿ |xᵢ| ≤ n . sup_1 ≤… 0C4K
Proposition 8 Let E be a normed space over a non-discrete valued division ring, let p be the norm on E, and let hatE be the additive topological group which is the completion of the additive gro… ↩ 1 0C4L
Proposition 9 Let E be a normed space over a non-discrete valued division ring K, and let H be a closed vector subspace of E. If, for each class dotx ∈ E/H, we put ||dotx|| = inf_x ∈ dotx ||x||… ↩ 1 0C4M
Proposition 10 Let (Eᵢ)_1 ≤ i ≤ n be a finite family of normed spaces over a non-discrete valued division ring K, and let E = ∏ᵢ₌₁ⁿ Eᵢ be the product vector space. If for each x = (xᵢ) ∈ E we put… ↩ 2 0C4N
Theorem 1 Let Eᵢ (1 ≤ i ≤ n) and F be normed spaces over a non-discrete valued division ring K, and let f be a multilinear mapping of ∏ᵢ₌₁ⁿ Eᵢ into F. Then f is continuous on ∏ᵢ₌₁ⁿ Eᵢ if and… ↩ 5 0C4O
Remark Two of the propositions proved earlier are consequences of this theorem: the continuity of the bilinear function tr, by virtue of the relation xr = t · x; and Proposition 7, by app… 0C4P
Definition 8 In a normed space E, a family (xᵢ) of points of E is said to be absolutely summable if the family (||xᵢ||) of norms of the xᵢ is summable in R. 0C4Q
Proposition 11 In a complete normed space E, every absolutely summable family is summable. ↩ 3 0C4R
Corollary In a complete normed space E, every absolutely convergent series is commutatively convergent. 0C4S
Definition 9 If A is an algebra over a non-discrete valued field K, a norm p(x) on A (A being considered as a vector space over K) is said to be compatible with the algebra structure of A if th… ↩ 2 0C4T
Example 1 Let K be a valued division ring and let K' be a subfield of the centre of K such that the trace on K' of the absolute value |x| of K is not the improper absolute value on K'. Then… 0C4U
Example 2 Let K be a non-discrete valued field and let Mₙ(K) be the ring of square matrices of order n over K. Regarded as a vector space over K, Mₙ(K) is isomorphic to K^n². If for each X =… 0C4V
Example 3 The set B(X; K) of all functions f on a set X with values in a non-discrete valued field K, such that x → |f(x)| is bounded on X, is an algebra over K; the norm |f| = sup_x ∈ X |f(… 0C4W
Proposition 12 If the series whose general term is zⁿ is convergent in A, then e - z is a unit of A and we have (e - z)⁻¹ = ∑ₙ₌₀^∞ zⁿ. Conversely, if ||z|| < 1 and if e - z is a unit in A, then t… 0C4X
Corollary Let A be a complete normed algebra. Then for each z ∈ A such that ||z|| < 1, e - z is a unit in A. ↩ 2 0C4Y
Proposition 13 Let G be the group of units of a complete normed algebra A. Then G is an open subset of A; the topology induced on G by the topology of A is compatible with the group structure of… ↩ 7 0C4Z
Proposition 14 In a complete valued division ring, the multiplicative group of non-zero elements is a complete group. 0C50
Remark Proposition 13 is not necessarily true for a non-complete normed algebra. For example, in the algebra C(I; R) of all finite continuous real-valued functions on I = [0, 1] [the norm… 0C51

§ 4. Normal spaces

Definition 1 A topological space X is said to be normal if it is Hausdorff and satisfies the following axiom: 0C5F
Theorem 1 (Urysohn) Axiom (Oᵥ) is equivalent to the following: ↩ 2 0C5G
Proposition 1 A compact space is normal. 0C5H
Proposition 2 A metrizable space is normal. ↩ 4 0C5I
Remark 1 Proposition 2 gives another necessary condition for metrizability; but this condition, even in conjunction with all the necessary conditions given in § 2, does not give a set of su… 0C5J
Remark 2 There are examples of normal spaces which are neither metrizable nor locally compact (see § 5, Exercise 16). 0C5K
Theorem 2 (Urysohn) Axiom (O_V) is equivalent to the following property: (O''_V) Given any closed subset A of X and any continuous real-valued function f (finite or not) defined on A, there exists an… ↩ 4 0C5L
Lemma 1 Let u be a continuous mapping of A into [ -i, +i ]; then there is a continuous mapping v of X into [ -i/3, +i/3 ], such that |u(x) - v(x)| ≤ 2/3 for all x ∈ A. 0C5M
Corollary If f is a finite continuous real-valued function defined on A, then there exists a finite continuous real-valued function g defined on X, which extends f. 0C5N
Remark If X is a normal space and if A is a closed subset of X, there exists also a continuous extension to X of every continuous mapping f of A into a cube K¹ (§ 1, no. 5); for we have t… 0C5O
Theorem 3 Let (Aᵢ)_i ∈ I be a locally finite open covering of a closed set Y in a normal space X. Then there is an open covering (Bᵢ)_i ∈ I of Y such that overlineBᵢ ⊂ Aᵢ for each i ∈ I. ↩ 5 0C5P
Remark Note that we have used only the fact that the covering (Aᵢ) is point-finite, i.e., that every point of X belongs to only a finite number of sets Aᵢ. 0C5Q
Definition 2 Let X be a topological space and let f be a real-valued function defined on X. The support of f, denoted by Supp(f), is the smallest closed set S in X such that f(x) = 0 for all x… 0C5R
Definition 3 Given a family (Aᵢ)_i ∈ I of subsets of a topological space X, a family (fᵢ)_i ∈ I of real-valued functions defined on X is said to be subordinate to the family (Aᵢ)_i ∈ I if we ha… 0C5S
Proposition 3 Given any locally finite open covering (Aᵢ)_i ∈ I of a normal space X, there exists a continuous partition of unity (fᵢ)_i ∈ I on X, subordinate to the covering (Aᵢ)_i ∈ I. 0C5T
Corollary Given any locally finite open covering (Aᵢ) of a closed set F in a normal space X, there exists a family (fᵢ) of continuous real-valued functions ≥ 0 on X, subordinate to the cover… 0C5U
Proposition 4 Every paracompact space is normal. ↩ 1 0C5V
Lemma 2 Let A, B be two disjoint closed subsets of a paracompact space X. If for each x ∈ A there is an open neighbourhood Vₓ of x and an open neighbourhood Wₓ of B which do not intersect… 0C5W
Corollary 1 Given any open covering (Aᵢ)_i ∈ I of a paracompact space X, there exists a continuous partition of unity (fᵢ)_i ∈ I on x, subordinate to the covering (Aᵢ). ↩ 3 0C5X
Corollary 2 If F is a closed subset of a paracompact space X, then every neighbourhood of F in X contains a closed (and therefore paracompact) neighbourhood of F. 0C5Y
Theorem 4 Every metrizable space is paracompact. ↩ 2 0C5Z
Lemma 3 Let R = (U_α)_α ∈ A be an open covering of a metrizable space X. Then there is a sequence (Gₙ) of locally finite families of open subsets of X, such that S = ⋃ₙ Sₙ is an open cover… ↩ 2 0C60
Lemma 4 Let (𝔖ₙ) be a sequence of locally finite families of open sets in a topological space X, such that 𝔖 = ⋃ₙ 𝔖ₙ is a covering of X. Then there exists a locally finite (but not necessa… 0C61
Lemma 5 Let X be a regular space such that, for each open covering R of X, there exists a (not necessarily open) locally finite covering U of X which refines R. Then for each open covering… ↩ 1 0C62
Lemma 6 Let X be a Hausdorff space such that, given any open covering ℜ of X, there exists a locally finite closed covering F of X which refines ℜ. Then X is paracompact. ↩ 2 0C63

§ 5. Baire spaces

Definition 1 A subset A of a topological space X is said to be nowhere dense if its closure has no interior points. 0C6Y
Example 1 The empty subset of X is nowhere dense. In a Hausdorff space X, a set consisting of a single point is nowhere dense if and only if the point is not isolated in X. A dense subset is… 0C6Z
Example 2 The frontier of an open or of a closed set is always nowhere dense. 0C70
Example 3 In the space Rⁿ, every linear subspace of dimension p < n is a nowhere dense set (Chapter VI, § 1, no. 4, Proposition 2). 0C71
Remark The frontier of an arbitrary subset need not be nowhere dense: for example, if A and overlineCA are both dense sets, then the frontier of A is the whole space. 0C72
Proposition 1 The union of a finite number of nowhere dense sets is nowhere dense. 0C73
Proposition 2 Let Y be a subspace of a topological space X, and let A be a subset of Y. If A is nowhere dense relative to Y, then A is nowhere dense relative to X. Conversely, if Y is open in X… ↩ 1 0C74
Definition 2 A subset A of a topological space X is said to be meagre if it is the union of a countable family of nowhere dense sets. ↩ 1 0C75
Definition 3 A topological space X is said to be a Baire space if it satisfies one or the other of the following two equivalent conditions: 0C76
Proposition 3 Every non-empty open subspace Y of a Baire space X is a Baire space. ↩ 1 0C77
Proposition 4 If every point of a topological space X has a neighbourhood which is a Baire space, then X is a Baire space. ↩ 1 0C78
Proposition 5 In a Baire space X, the complement of a meagre set is a Baire space. 0C79
Theorem 1 (Baire) (i) Every locally compact space X is a Baire space. (ii) Every topological space X on which there exists a metric, compatible with the topology of X and which defines on X the stru… ↩ 4 0C7A
Remark There are Baire spaces which belong to neither of these two categories, in particular Baire spaces which are neither metrizable nor locally compact (Exercise 16); there are also me… 0C7B
Theorem 2 Let X be a Baire space and let (f_α) be a family of lower semi-continuous real-valued functions on X such that, at every point x of X, the upper envelope sup_α f_α(x) is finite. Th… ↩ 1 0C7C
Remark The theorem may be false if we do not suppose that X is a Baire space. For example if, for each rational number p/q (p, q being coprime integers, q > 0) we put f(p/q) = q, we have… 0C7D

§ 6. Polish spaces, Souslin spaces, Borel sets

Definition 1 A topological space X is said to be Polish if it is metrizable of countable type (§ 2, no. 8) and if there is a metric, compatible with the topology of X, with respect to which X i… 0C88
Proposition 1 a) Every closed subspace of a Polish space is Polish. b) The product of a countable family of Polish spaces is Polish. c) The sum of a countable family of Polish spaces is Polish. ↩ 6 0C89
Proposition 2 Every open subspace of a Polish space is Polish. 0C8A
Corollary Every locally compact, metrizable space X is Polish. 0C8B
Proposition 3 Let X be a Hausdorff topological space. Then the intersection of a sequence (Aₙ) of Polish subspaces of X is a Polish subspace. ↩ 2 0C8C
Lemma 1 Let (Aₙ) be a sequence of subsets of a Hausdorff topological space X. Then the restriction of the diagonal mapping f : X → Xᴺ to the subspace ⋂ₙ Aₙ of X is a homeomorphism of ⋂ₙ Aₙ… ↩ 2 0C8D
Corollary The set of irrational numbers, endowed with the topology induced by that of the real line R, is a Polish space. ↩ 1 0C8E
Theorem 1 A subspace Y of a Polish space X is Polish if and only if Y is the intersection of a countable family of open sets in X. ↩ 4 0C8F
Corollary 1 A space X is Polish if and only if it is homeomorphic to a countable intersection of open sets in the cube I^N, where I is the interval [0, 1] of R. ↩ 2 0C8G
Corollary 2 Let X and Y be two Polish spaces and let f : X → Y be a continuous mapping. If Z is a Polish subspace of Y, then overlinef⁻¹(Z) is a Polish subspace of X. 0C8H
Definition 2 A topological space X is said to be a Souslin space if it is metrizable and if there exists a Polish space P and a continuous mapping of P onto X. A subset A of a topological space… ↩ 2 0C8I
Proposition 4 Every Souslin space X is of countable type. 0C8J
Proposition 5 Every closed (resp. open) subspace of a Souslin space X is Souslin. ↩ 1 0C8K
Proposition 6 Let X be a Souslin space, let Y be a Hausdorff space and let f : X → Y be a continuous mapping. Then the inverse image under f of a Souslin subspace A of Y is a Souslin subspace of… 0C8L
Proposition 7 The product and the sum of a countable family of Souslin spaces are Souslin spaces. 0C8M
Proposition 8 Let X be a metrizable space and let (Aₙ) be a sequence of Souslin subspaces of X. Then the union and the intersection of the Aₙ are Souslin spaces. ↩ 2 0C8N
Proposition 9 Let X be a metrizable space, and let A be a relatively compact Souslin subspace of X. Then there exists a compact metrizable space K, a decreasing sequence (Bₙ) of subsets of K, ea… ↩ 1 0C8O
Definition 3 Let X be a set and let 𝔏 be a set of subsets of X. 𝔏 is said to be a σ-algebra on X if the following conditions are satisfied: 0C8P
Definition 4 In a topological space X, the elements of the σ-algebra generated by the set of all closed subsets of X are called Borel sets in X. 0C8Q
Proposition 10 Let f be a continuous mapping of a topological space X into a topological space Y. Then the inverse image under f of every Borel set in Y is a Borel set in X. 0C8R
Proposition 11 In a Souslin space X, every Borel set is a Souslin set. 0C8S
Corollary Let f be a continuous mapping of a Souslin space X into a metrizable space Y. If B is any Borel set in X, then f(B) is a Souslin set in Y. 0C8T
Remark 1 Even when X and Y are Polish spaces it is not in general true that the image of a Borel set in X under a continuous mapping of X into Y is a Borel set in Y (cf. Exercise 6; and no… 0C8U
Remark 2 Let X be a topological space and let Y be a Borel subset of X. Then the Borel sets of the space Y are exactly the Borel sets of X which are contained in Y. For (i) the Borel sets i… 0C8V
Definition 5 A topological space is said to be zero-dimensional if it is Hausdorff and if every point has a fundamental system of neighbourhoods which are both open and closed. ↩ 1 0C8W
Definition 6 A topological space X is a Lusin space if it is metrizable and if there exists a zero-dimensional Polish space P and a continuous bijective mapping of P onto X. ↩ 2 0C8X
Proposition 12 A metrizable space is a Lusin space if and only if it is the image of a Polish space under a continuous bijective mapping. ↩ 1 0C8Y
Lemma 2 If a metrizable space X is such that there exists a countable partition (Aₙ) of X formed of Lusin subspaces, then X is a Lusin space. ↩ 1 0C8Z
Definition 7 A sieve is a sequence C = (Cn, pn){n \geq 0} such that, for each n, Cn is a countable set and pn is a surjection of C{n+1} onto Cn. 0C90
Lemma 3 Every metric space X of countable type possesses a sifting. If also X is zero-dimensional, then X has a strict sifting. ↩ 2 0C91
Proposition 13 If X is any Lusin (resp. Souslin) space, there exists a sieve C and a continuous bijection (resp. surjection) of L(C) onto X. ↩ 4 0C92
Theorem 2 Let X be a metrizable space. If we are given a sequence (Xₙ) of mutually disjoint Souslin subspaces of X, then there exists a sequence (Bₙ) of mutually disjoint Borel sets in X suc… ↩ 5 0C93
Lemma 4 Let (Aₙ), (A'ₘ) be two sequences of subsets of a topological space X. Suppose that for each pair (Aₙ, A'ₘ) there is a Borel set Bₙₘ of X such that Bₙₘ ⊃ Aₙ and Bₙₘ ∩ A'ₘ = ∅. Then… ↩ 1 0C94
Lemma 5 Let X be a Hausdorff space and let A, A' be two disjoint Souslin subspaces of X. Then there is a Borel set B of X such that B ⊃ A and B ∩ A' = ∅. ↩ 1 0C95
Corollary If a countable partition of a metrizable space is formed of Souslin sets, then these sets are Borel sets. In particular, every Souslin set in a metrizable space, whose complement i… 0C96
Theorem 3 Let X be a Lusin space. Then a subspace of X is a Lusin space if and only if it is a Borel set. ↩ 2 0C97
Lemma 6 In a Lusin space X, every Borel set is a Lusin subspace of X. 0C98
Lemma 7 Every Lusin subspace A of a metrizable space X is a Borel set in X. 0C99
Corollary If f is a continuous injective mapping of a Lusin space (or, in particular, a Polish space) X into a metrizable space Y, then f(X) is a Borel set in Y. 0C9A
Theorem 4 Let X be a Polish space and let R be an equivalence relation on X, such that the equivalence classes mod R are closed in X and the saturation (with respect to R) of each closed set… ↩ 2 0C9B
Remark In particular a closed equivalence relation R satisfies the hypotheses of Theorem 4. When X is a compact metrizable space, Theorem 4 therefore applies to any Hausdorff equivalence… 0C9C
Definition 8 Let X be a Hausdorff topological space. A capacity on X is a mapping f of the set \mathscr{P}(X) of all subsets of X into the extended real line \overline{\mathbf{R}}, satisfying t… ↩ 1 0C9D
Example \ Let \mu be a positive measure on a locally compact space X; then the corresponding outer measure \mu^ is a capacity on X. It can be shown that in Euclidean space \mathbf{R}^n (n… 0C9E
Definition 9 Let f be a capacity on X; a subset A of X is said to be capacitable (with respect to f) if f(A) = supₖ f(K), where K runs through the set of compact subsets of A. 0C9F
Proposition 14 Let K be a compact space and let f be a capacity on K. If A is the intersection of a decreasing sequence (Aₙ) of subsets of K, each of which is a countable union of closed sets, th… ↩ 1 0C9G
Theorem 5 Let X be a metrizable space and let Y be a relatively compact Souslin subspace of X. Then Y is capacitable with respect to every capacity f on X. ↩ 1 0C9H
Proposition 15 Let φ be a continuous mapping of a Hausdorff space K into a Hausdorff space X, and let f be a capacity on X. If for each subset A of K we put g(A) = f(φ(A)), then g is a capacity o… ↩ 1 0C9I
Remark \ If μ is a positive measure on a locally compact metrizable space X, then every Souslin subset A of X is μ-measurable. For if K is any compact subset of X, K ∩ A is a relatively c… 0C9J

App. 0. Infinite products in normed algebras

Definition 1 The sequence (xₙ)_n ∈ N is said to be multipliable in the normed algebra A if the mapping J → p_J has a limit with respect to the filter of sections of the set 𝔉(N) of finite subse… ↩ 3 0CA3
Remark 1 When A is a commutative algebra, Definition 1 is identical with that given in Chapter III, § 5, no. 1, Remark 3; but when A is not commutative, the order structure of the index set… 0CA4
Remark 2 Definition 1 can be immediately generalized to the case of a family (xn){n \in I} whose index set I is a subset of \mathbf{Z} (linearly ordered by the order induced by that of \mat… 0CA5
Theorem 1 Let (xn){n \in \mathbf{N}} be a sequence of points in a complete normed algebra A. a) If (xn) is multipliable and if its product is a unit of A, then for each ε > 0 there exists a… ↩ 5 0CA6
Remark If (xₙ) is multipliable and its product is not a unit, the condition of Theorem 1 is not necessarily satisfied: for example, if all the xₙ are equal to the same element x, where |x… 0CA7
Corollary 1 If (xₙ) is a multipliable sequence whose product is a unit of A, then lim_n → ∞ xₙ = e. 0CA8
Corollary 2 If (xₙ) is a multipliable sequence whose product is a unit of A, then every subsequence (x_nₖ)_k ∈ N of (xₙ) [(nₖ) being a strictly increasing sequence of integers] is multipliable… 0CA9
Theorem 2 Let A be a complete normed algebra. If (uₙ) is an absolutely convergent series of elements of A, then the sequence (e + uₙ) is multipliable in A; and if all the elements e + uₙ are… ↩ 1 0CAA
Corollary If the series whose general term is uₙ is absolutely convergent, and if none of the elements e - uₙ is a zero divisor in A, then the product ∏_n ∈ N (e + uₙ) is not a zero divisor… 0CAB
Proposition 1 Let A be a normed algebra of finite rank over R. If (e + uₙ) is a multipliable sequence in A, whose product is a unit of A, then the series whose general term is uₙ is absolutely c… ↩ 1 0CAC
Lemma 1 For each finite subset I of N, let φ(I) = sup_J ⊂ I ||p_J - e||. Then for each subset J of I we have 0CAD
Lemma 2 If I is a finite subset of N such that φ(I) < 1/c, then 0CAE
Proposition 2 Let (xₙ) be a sequence of points in a complete normed algebra A. ↩ 1 0CAF
Corollary 1 If the infinite product with general factor xₙ is convergent, and if ∏ₙ₌₀^∞ xₙ is a unit, then lim_n → ∞ xₙ = e. 0CAG
Corollary 2 If the infinite product with general factor xₙ is convergent, and if ∏ₙ₌₀^∞ xₙ is a unit, then the infinite product with general factor yₙ = xₙ₊ₕ (n ≥ 0) is convergent. 0CAH
Proposition 3 Let (kₙ) be a strictly increasing sequence of integers ≥ 0, such that k₀ = 0; if the infinite product with general factor xₙ converges, and if we put 0CAI

X. Function spaces

§ 1. The uniformity of $\mathscr{G}$-convergence

Definition 1 The uniformity on the set F(X; Y) which has as a fundamental system of entourages the set of subsets W(V), where V runs through the set of entourages of Y, is called the uniformity… 0CAR
Definition 2 Let X be a set, Y a uniform space, 𝔖 a set of subsets of X. The uniformity of uniform convergence in the sets of 𝔖, or simply the uniformity of 𝔖-convergence, is the coarsest unifo… 0CAS
Remark 1 The general definition of the entourages of an initial uniformity (Chapter II, § 2, no. 3, Proposition 4) shows that a fundamental system of entourages of F_S(X; Y) may be obtained… ↩ 1 0CAT
Remark 2 However, the uniformity of S-convergence is unaltered by replacing S by the set S' of all subsets of X which are contained in finite unions of sets of S. In the study of S-converge… ↩ 1 0CAU
Remark 3 The uniformity of S-convergence is the inverse image, under the mapping u → (u|A)_A ∈ S of F(X; Y) into ∏_A ∈ S Fᵤ(A; Y), of the uniformity of this product space (Chapter II, § 2… ↩ 1 0CAV
Proposition 1 If Y is Hausdorff and S is a covering of X, then the space F_S(X; Y) is Hausdorff. ↩ 3 0CAW
Remark 4 Let H be a subset of F(X; Y). By abuse of language, the uniformity (resp. topology) induced on H by the uniformity (resp. topology) of S-convergence on F(X; Y) is called the unifor… 0CAX
Remark 5 Let L be a set filtered by a filter 𝔖, and let λ → u_λ be a mapping of L into F_S(X; Y) which has a limit v with respect to 𝔖; we say then that, with respect to the filter 𝔖, the m… ↩ 1 0CAY
Remark 6 It follows immediately from Definitions 1 and 2 that, for every x ∈ ⋃_A ∈ G A, the mapping u → u(x) of F_G(X; Y) into Y is uniformly continuous. Hence, in particular, if overlineH… ↩ 8 0CAZ
Proposition 2 Let X₁, X₂ be two sets, let Y be a uniform space and let Sᵢ be a set of subsets of Xᵢ (i = 1, 2) and S₁ × S₂ the set of subsets of X₁ × X₂ of the form A₁ × A₂, where Aᵢ ∈ Sᵢ, i = 1… ↩ 1 0CB0
Proposition 3 a) Let X be a set; let S be a set of subsets of X; let Y, Y' be two uniform spaces; and let f : Y → Y' be a uniformly continuous mapping. Then the mapping u → f ∘ u of F_S(X; Y) in… 0CB1
Proposition 4 Let X, Y be two sets, let (X_λ)_λ ∈ L be a family of sets and let (Y_μ)_μ ∈ M be a family of uniform spaces. For each λ ∈ L, let S_λ be a set of subsets of X_λ, let g_λ be a mappin… ↩ 1 0CB2
Corollary Let X be a set, let (Yᵢ)_i ∈ I be a family of uniform spaces and let S be a set of subsets of X. If we endow ∏_i ∈ I Yᵢ with the product uniformity, the canonical bijection of the… 0CB3
Proposition 5 Let Φ be a set, Y a uniform space and S a set of subsets of X. Then a filter Φ on F_S(X; Y) converges to u₀ if and only if Φ is a Cauchy filter with respect to the uniformity of S-… ↩ 3 0CB4
Corollary 1 A subspace H of F_S(X; Y) is complete if and only if, for each Cauchy filter Φ on H, there exists u₀ ∈ H such that Φ converges pointwise to u₀ in B = ⋃_A ∈ S A. ↩ 1 0CB5
Corollary 2 Let S₁, S₂ be two sets of subsets of X, whose union is the same and which are such that S₁ ⊂ S₂, and let H be a subset of F(X; Y). Then if H is complete with respect to S₁-converge… ↩ 1 0CB6
Corollary 3 Let H be a subset of F(X; Y) such that, for each x ∉ B = ⋃_A ∈ G A, the closure of H(x) in Y is a complete subspace of Y. Then the closure overlineH of H in F_G(X; Y) is a complete… 0CB7
Theorem 1 Let X be a set, let G be a set of subsets of X, and let Y be a complete uniform space. Then the uniform space F_G(X; Y) is complete. ↩ 2 0CB8
Proposition 6 Let X be a topological space, Y a uniform space and G a set of subsets of X. For each A ∈ G and each closed entourage V of Y, the traces on C(X; Y) × C(X; Y) of W(A, V) and W(overl… ↩ 2 0CB9
Corollary Let B be a dense subset of X. On C(X; Y), the structure of uniform convergence is identical with the structure of uniform convergence in B. ↩ 1 0CBA
Proposition 7 Let X be a topological space, let 𝔖 be a set of subsets of X and let Y be a uniform space. If Y is Hausdorff and if the union B of the sets of 𝔖 is dense in X, then C_𝔖(X; Y) is Ha… 0CBB
Proposition 8 Let X be a set, 𝔗 a filter on X, and let Y be a uniform space. Then the set H of mappings u : X → Y such that u(𝔗) is a Cauchy filter base on Y is closed in Fᵤ(X; Y). ↩ 3 0CBC
Corollary 1 Let X be a topological space and Y a uniform space. The set of mappings of X into Y which are continuous at a point x₀ ∈ X is closed in Fᵤ(X; U). ↩ 1 0CBD
Corollary 2 Let X, L be two sets filtered by filters 𝔗, 𝔊 respectively, and let Y be a complete uniform space. For each λ ∈ L, let u_λ be a mapping of X into Y. Suppose that (i) the family (u_… 0CBE
Theorem 2 Let X be a topological space, Y a uniform space. Then the set C(X; Y) of continuous mappings of X into Y is a closed subset of the space F(X; Y) endowed with the topology of unifor… ↩ 4 0CBF
Corollary 1 If Y is a complete uniform space, then Cᵤ(X; Y) is complete. 0CBG
Corollary 2 Let X be a topological space, S a set of subsets of X, and Y a uniform space. Let tildeC_S(X; Y) denote the set of all mappings of X into Y whose restriction to each set of S is co… ↩ 2 0CBH
Corollary 3 Let X be a topological space which is either metrizable or locally compact, and let Y be a uniform space. Then C(X; Y) is closed in the uniform space F_c(X; Y); if in addition Y is… ↩ 1 0CBI
Remark 1 In general, the set C(X; Y) is not closed in F(X; Y) with respect to the topology of pointwise convergence: in other words, a pointwise limit of continuous functions is not necessa… 0CBJ
Remark 2 A filter on C(X; Y) can converge pointwise to a continuous function without converging uniformly to this function. ↩ 1 0CBK
Remark 3 If X is a uniform space, a proof analogous to that of Proposition 8 shows that the set of uniformly continuous mappings of X into Y is closed in Fᵤ(X; Y). 0CBL
Remark 4 Suppose that the uniform space Y carries a commutative and associative law of composition, written additively, such that the mapping (y, y') → y + y' is continuous on Y × Y. Then… 0CBM
Proposition 9 Let X be a topological space, Y a uniform space. Then the mapping (f, x) → f(x) of Cᵤ(X; Y) × X into Y is continuous. ↩ 1 0CBN

§ 2. Equicontinuous sets

Definition 1 Let X be a topological space and Y a uniform space. A subset H of F(X; Y) is said to be equicontinuous at a point x₀ ∈ X if, for each entourage V of Y, there is a neighbourhood U o… 0CC0
Definition 2 Let X and Y be two uniform spaces. A subset H of F(X; Y) is said to be uniformly equicontinuous if, for each entourage V of Y, there exists an entourage U of X such that we have (f… 0CC1
Example 1 Let X be a topological space (resp. a uniform space) and Y a uniform space. Every finite set of continuous (resp. uniformly continuous) mappings of X into Y is equicontinuous (resp… 0CC2
Example 2 Let X, Y be two metric spaces, d (resp. d') the metric on X (resp. Y), and let k, α be two real numbers 0. Then the set of all mappings f : X → Y such that d'(f(x), f(x')) ≤ k(d(x… 0CC3
Example 3 Let G be a topological group, let Y be a uniform space and let f : G → Y be a uniformly continuous mapping [G being endowed with its left uniformity (Chapter III, § 3, no. 1)]. For… 0CC4
Proposition 1 Let T be a set, let S be a set of subsets of T, let Y be a uniform space, X a topological (resp. uniform) space, and let f be a mapping of T × X into Y. For each A ∈ S, let H_A ⊂ F… ↩ 5 0CC5
Corollary 1 Let X be a topological (resp. uniform) space, Y a uniform space and H a subset of F(X; Y). For each x ∈ X, let tildex denote the mapping h → h(x) of H into Y. Then H is equicontinu… ↩ 1 0CC6
Corollary 2 Let X be a compact space, Y a uniform space. Then every equicontinuous subset of F(X; Y) is uniformly equicontinuous. ↩ 2 0CC7
Corollary 3 Let T, X be topological spaces, let Y be a uniform space and let f be a mapping of T × X into Y. Then f is continuous, provided that the following conditions are satisfied: 0CC8
Corollary 4 Let X be a topological space, Y a uniform space, H an equicontinuous set of mappings of X into Y. If H is endowed with the topology of pointwise convergence, then the mapping (h, x… ↩ 1 0CC9
Corollary 5 Let X be a topological space, let Y, Z be two uniform spaces and let H be an equicontinuous set of mappings of Y into Z. If H, C(X; Y) and C(X; Z) are endowed with the topology of… ↩ 1 0CCA
Proposition 2 Let T, X, Y be uniform spaces and let f be a mapping of T × X into Y. Then f is uniformly continuous if and only if the following two conditions are satisfied: 0CCB
Corollary Let X, Y be two uniform spaces and let H be a subset of F(X; Y). Then H is uniformly equicontinuous if and only if the mapping (h, x) → h(x) of H × X into Y is uniformly continuous… 0CCC
Proposition 3 Let X be a topological (resp. uniform) space, let (Yᵢ)_i ∈ I be a family of uniform spaces, let Y be a set, and for each i ∈ I, let fᵢ be a mapping of Y into Yᵢ. Let Y be endowed w… ↩ 1 0CCD
Proposition 4 Let X, Y be two uniform spaces and let H be a set of uniformly continuous mappings of X into Y. Let hatX, hatY be the Hausdorff completions of X, Y respectively, and let tildeH den… ↩ 1 0CCE
Proposition 5 Let G, G' be two topological groups endowed with their left uniformities, and let H be a set of homomorphisms of G into G'. Then the following conditions are equivalent: a) H is eq… 0CCF
Proposition 6 Let X be a topological (resp. uniform) space, let Y be a uniform space and let H be a subset of F(X; Y). Then H is equicontinuous at a point x₀ ∈ X (resp. uniformly equicontinuous)… ↩ 2 0CCG
Theorem 1 Let X be a topological (resp. uniform) space, let Y be a uniform space and let H be an equicontinuous (resp. uniformly equicontinuous) subset of C(X; Y). Then the following uniform… ↩ 2 0CCH
Corollary Under the hypotheses of Theorem 1, the closure overlineH of H in F(X; Y) with respect to the topology of pointwise convergence is the same as the closure of H in C(X; Y) with respe… 0CCI
Theorem 2 (Ascoli) Let X be a topological (resp. uniform) space, let S be a covering of X, let Y be a uniform space and H a set of mappings of X into Y such that, for each A ∈ S and each u ∈ H, the r… ↩ 4 0CCJ
Corollary 1 Let X be a topological (resp. uniform) space, let Y be a Hausdorff uniform space and let H be an equicontinuous (resp. uniformly equicontinuous) subset of C(X; Y). Suppose that H(x… ↩ 2 0CCK
Corollary 2 Let X be a topological (resp. uniform) space, let Y be a complete Hausdorff uniform space and let H be an equicontinuous (resp. uniformly equicontinuous) subset of C(X; Y). Suppose… 0CCL
Corollary 3 Let X be a locally compact space, Y a Hausdorff uniform space, H a subset of C(X; Y). Then H is relatively compact in C_c(X; Y) if and only if H is equicontinuous and H(x) relative… ↩ 1 0CCM
Remark Let X be a topological space, Y a uniform space and S a set of subsets of X. Then on every precompact subset H of F_S(X; Y), the uniformity of S-convergence is the same as the unif… 0CCN

§ 3. Special function spaces

Proposition 1 If Y is a metrizable uniform space, the uniformity of uniform convergence on F(X; Y) is metrizable. 0CD8
Corollary Let X be a topological space and let Y be a metrizable uniform space. Suppose that there is a sequence (Kₙ) of compact subsets of X such that every compact subset of X is contained… 0CD9
Proposition 2 Let X be a set and Y a metric space. The set B(X; Y) of bounded mappings is both open and closed in the space Fᵤ(X; Y). ↩ 3 0CDA
Corollary 1 Let X be a set and Y a metric space. Then B_S(X; Y) is closed in F_S(X; Y). In particular, if Y is complete then B_S(X; Y) is complete with respect to the uniformity of S-convergen… ↩ 2 0CDB
Corollary 2 Let X be a topological space and Y a metric space. Then the space of all bounded continuous mappings of X into Y is both open and closed in Cᵤ(X; Y); it is complete if Y is complet… 0CDC
Proposition 3 If the normed space Y is complete, then every series (uₙ) of bounded mappings of X into Y which is absolutely convergent in the normed space B(X; Y) (i.e. which is such that ∑ₙ₌₀^∞… 0CDD
Remark 1 If ∑ₙ₌₀^∞ |uₙ| < +∞, then ∑ₙ₌₀^∞ |uₙ(x)| ≤ ∑ₙ₌₀^∞ |uₙ| < +∞ for each x ∈ X; in other words, for each x ∈ X the series with general term uₙ(x) is absolutely convergent in the space… 0CDE
Proposition 4 Let Xᵢ (1 ≤ i ≤ n) and Y be normed vector spaces over a non-discrete valued division ring K, and let X = ∏ᵢ₌₁ⁿ Xᵢ. Then the set of all multilinear mappings of X into Y is closed in… ↩ 2 0CDF
Proposition 5 Under the hypotheses of Proposition 4, the set L(X₁, …, Xₙ; Y) of continuous multilinear mappings of X into Y is closed in F(X; Y) with respect to the topology of bounded convergen… 0CDG
Proposition 6 The multilinear mapping (u, x₁, …, xₙ) → u(x₁, …, xₙ) of the normed space L(X₁, …, Xₙ; Y) × X₁ × ⋯ × Xₙ into Y is continuous. 0CDH
Proposition 7 Let X, Y, Z be three normed spaces over K. The canonical mapping of the normed space L(X, Y; Z) into the space of linear mappings of X into L(Y; Z) which sends each u ∈ L(X, Y; Z)… 0CDI
Proposition 8 Let X, Y, Z be three normed spaces over K. The bilinear mapping (u, v) → v ∘ u of L(X; Y) × L(Y; Z) into L(X; Z) is continuous. 0CDJ
Remark 2 The set L(Rᵐ; Rⁿ) of linear (necessarily continuous) mappings of Rᵐ into Rⁿ can be identified with the set M_n,m(R) of matrices with n rows and m columns with coefficients in R and… 0CDK
Theorem 1 Let X be a compact space. a) If X is metrizable and if Y is any metrizable uniform space of countable type (Chapter IX, § 2, no. 8), then the metrizable space Cᵤ(X; Y) of continuou… ↩ 1 0CDL
Corollary Let X be a locally compact space whose topology admits a countable base, and let Y be a metrizable uniform space of countable type. 0CDM
Theorem 2 Let X be a topological space, Y a uniform space. For each compact subset K of X and each open subset U of Y, let T(K, U) denote the set of all continuous mappings u : X → Y such th… ↩ 2 0CDN
Definition 1 Let X, Y be two topological spaces, not necessarily uniformizable. For each compact subset K of X and each open subset U of Y, let T(K, U) be the set of all u ∈ C(X; Y) such that u… 0CDO
Example Let I be the interval [0, 1] in R. If Y is any topological space, the space C_c(I; Y) is called the space of paths in Y. For each y ∈ Y, the subspace Ω_y(Y) of C_c(I; Y) consisting… 0CDP
Remark 1 Likewise, the topology induced on C(X; Y) by the product topology on Y^X = F(X; Y) is called the topology of pointwise convergence (Y being not necessarily uniformizable); it is ge… 0CDQ
Remark 2 Let S be a subbase of the topology of Y, and let K be a set of compact subsets of X with the following property: ↩ 1 0CDR
Theorem 3 Let X, Y, Z be three topological spaces and let f be a mapping of X × Y into Z. If f is continuous then tildef: x → f(x, .) is a continuous mapping of X into C_c(Y; Z). The convers… ↩ 5 0CDS
Corollary 1 Let X be a locally compact space, Y a topological space, H a subset of C(X; Y). Then the compact-open topology on H is the coarsest for which the mapping (u, x) → u(x) of H × X int… ↩ 5 0CDT
Remark 3 Let X be a locally compact space and Y a Hausdorff topological space. If T is a topology on a subset H of C(X; Y) such that the mapping (u, x) → u(x) is continuous on H × X and if… ↩ 2 0CDU
Corollary 2 Let X, Y, Z be three topological spaces such that X is Hausdorff and Y is locally compact. Then the restriction to C(X × Y; Z) of the canonical bijection F(X × Y; Z) → F(X; F(Y; Z)… 0CDV
Proposition 9 Let X, Y, Z be three topological spaces, Y being locally compact. Then the mapping (u, v) → v ∘ u of C_c(X; Y) × C_c(Y; Z) into C_c(X; Z) is continuous. ↩ 2 0CDW
Proposition 10 Let X be a uniform space and let H be an equicontinuous set of homeomorphisms of X onto itself. If H and H⁻¹ are endowed with the topology of pointwise convergence in X, then the m… ↩ 1 0CDX
Corollary Let X be a uniform space and let H be an equicontinuous group of homeomorphisms of X onto itself. Then the topology of pointwise convergence in X is compatible with the group struc… ↩ 1 0CDY
Proposition 11 Let X be a compact space and let Γ be the group of all homeomorphisms of X onto itself. Then the topology of uniform convergence in X is compatible with the group structure of Γ. ↩ 1 0CDZ
Proposition 12 On the group Γ of all homeomorphisms of a locally compact space X, the topology T_β is the coarsest for which the mappings u → u and u → u⁻¹ of Γ into C_c(X; X) are continuous. ↩ 2 0CE0
Theorem 4 Let G be a subgroup of the group Γ of all homeomorphisms of a locally compact space X. Suppose that, in the space C_c(X; X), there is a neighbourhood V of the identity mapping e su… ↩ 2 0CE1
Corollary Let G be a group of homeomorphisms of a locally compact space X. If the closure overlineG of G in C_c(X; X) is compact, then overlineG is a group of homeomorphisms of X, and the to… ↩ 1 0CE2

§ 4. Approximation of continuous real-valued functions

Corollary Let (uₙ) be an increasing (resp. decreasing) sequence of continuous real-valued functions on X. If the upper (resp. lower) envelope f of the sequence (uₙ) is finite and continuous… 0CEW
Proposition 2 Let X be a compact space, and let H be a set of continuous real-valued functions on X such that, given any two functions u ∈ H, v ∈ H, the functions sup(u, v) and inf(u, v) are in… ↩ 2 0CEX
Remark When the set H satisfies the conditions of Proposition 2, it is a lattice with respect to the ordering f ≤ g. But it should be remarked that a subset H of C can be a lattice with r… 0CEY
Corollary Suppose that H is such that, whenever u ∈ H and v ∈ H, we have sup(u, v) ∈ H and inf(u, v) ∈ H; and is such that, given any two distinct points x, y of X and any two real numbers α… ↩ 1 0CEZ
Definition 1 If X is any set, a set H of mappings of X into a set Y is said to separate the elements of a subset A of X (or to be a separating set for the elements of A) if, given any two disti… 0CF0
Theorem 2 (Stone) Let X be a compact space, and let H be a vector subspace of C(X; R) such that 1) the constant functions belong to H; 2) if u ∈ H, then |u| ∈ H; 3) H separates the points of X. Then… ↩ 2 0CF1
Theorem 3 (Weierstrass-Stone) Let X be a compact space and let H be a set of continuous real-valued functions on X which separates the points of X. Then every continuous real-valued function on X can be uniform… ↩ 5 0CF2
Lemma 1 For each real number ε > 0 and each compact interval I ⊂ R there exists a polynomial p(t) with no constant term such that |p(t) - |t|| ≤ ε for all t ∈ I. 0CF3
Lemma 2 Let (Pₙ) be the sequence of polynomials without constant terms defined by (1) p₀(t) = 0, pₙ₊₁(t) = pₙ(t) + 1/2 (t - (pₙ(t))²), n ≥ 0. In the interval [0, 1] the sequence (pₙ) is in… 0CF4
Proposition 3 Let (Kᵢ)_i ∈ I be a family of compact intervals of R, K = ∏_i ∈ I Kᵢ their product, and let X be a compact subspace of K. Then every continuous real-valued function on X can be uni… 0CF5
Proposition 4 Let X be a compact space, let A be a closed subspace of X, and let H be a set of continuous real-valued functions on X which separates the points of C_A and is such that A is the i… ↩ 2 0CF6
Theorem 4 Let (Xᵢ)_i ∈ I be a family of compact spaces, and let X = ∏_i ∈ I Xᵢ. Then every continuous real-valued function on X can be uniformly approximated by sums of a finite number of fu… 0CF7
Proposition 5 Let X be a compact space, Y a normed space over R and H a subset of C(X; R). If every continuous real-valued function on X can be uniformly approximated by functions of H, then eve… 0CF8
Proposition 6 Let X be a compact space and let H be a set of continuous real-valued functions on X which separates the points of X. Then every continuous mapping of X into a normed space Y over… ↩ 1 0CF9
Proposition 7 Let X be a compact space and let H be a set of continuous complex-valued functions on X which separates the points of X. Then every continuous mapping of X into a normed space Y ov… ↩ 2 0CFA
Corollary 1 If X is a compact subset of the space Cⁿ, then every continuous mapping (z₁, z₂, …, zₙ) → f(z₁, z₂, …, zₙ) of X into a normed space Y over the field C can be uniformly approximated… 0CFB
Corollary 2 Let X be a locally compact space and let C₀(X) be the normed C-algebra of continuous mappings of X into C which tend to 0 at infinity. Let A be a subalgebra of C₀(X) which separate… 0CFC
Proposition 8 Let P be the set of all periodic continuous mappings of Rᵐ into C whose group of periods contains Zᵐ. Then every function belonging to P can be uniformly approximated in Rᵐ by line… 0CFD