§ 1. Uniform spaces · General Topology

1. Uniform spaces

1. Definition of a uniform structure

Definition 1

A uniform structure* (or uniformity) on a set $X$ is a structure given by a set $\mathfrak{U}$ of subsets of $X \times X$ which satisfies axioms $(\mathrm{F}_1)$ and $(\mathrm{F}_{\mathrm{II}})$ of Chapter I, § 6, no. 1 and also satisfies the following axioms:

(U₁) Every set belonging to $\mathfrak{U}$ contains the diagonal $\Delta$ .

(U₂) If $V \in \mathfrak{U}$ then $\overline{V}^{-1} \in \mathfrak{U}$ .

(U₃) For each $V \in \mathfrak{U}$ there exists $W \in \mathfrak{U}$ such that $W \circ W \subset V$ .

!Figure 2

The sets of $\mathfrak{U}$ are called entourages of the uniformity defined on $X$ by $\mathfrak{U}$ . A set endowed with a uniformity is called a uniform space.

If $V$ is an entourage of a uniformity on $X$ , we may express the relation $(x, x') \in V$ by saying that " $x$ and $x'$ are $V$ -close".

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 a relation

R $\{ x, y \}$ is true whenever $x$ and $y$ are close enough if there is an entourage $V$ such that the relation $(x, y) \in V$ implies $R \{ x, y \}$ .

Remark 2

The conjunction of axioms (U_{II}) and (U_{III}) is equivalent (assuming the other axioms of uniform structures) to the following axiom:

(U_a) For each $V \in \mathfrak{U}$ there exists $W \in \mathfrak{U}$ such that $W \circ \overline{W} \subset V$ (*).

Clearly (U_{II}) and (U_{III}) imply (U_a). Conversely, if (U_a) is satisfied we have $\overline{W} = \Delta \circ W \subset V$ , by (U_I); hence $W \subset \overline{V}$ and therefore [by (F_1)] $\overline{V} \in \mathfrak{U}$ . Let $W' = W \cap \overline{W}$ ; then $W' \in \mathfrak{U}$ by what has just been proved and axiom (F_{II}), and we have $W' \circ W' \subset W \circ \overline{W} \subset V$ .

Throughout this chapter we shall write $\overline{V}^2$ instead of $V \circ V$ , and in general $\overline{V}^{n-1} = \overline{V} \circ V = V \circ \overline{V}^{n-1}$ , for each integer $n > 1$ and each subset $V$ of $X \times X$ .

Remark 3

If $X$ is not empty, then axiom (U_I) implies that no set of $\mathfrak{U}$ is empty, and therefore $\mathfrak{U}$ is a filter on $X \times X$ . There is only one uniformity on the empty set, namely $\mathfrak{U} = \{ \emptyset \}$ .

Definition 2

A fundamental system of entourages of a uniformity is any set $\mathfrak{B}$ of entourages such that every entourage contains a set belonging to $\mathfrak{B}$ .

Axiom (U_{III}) shows that if $n$ is any integer $> 0$ and $V$ runs through a fundamental system of entourages, then the sets $\overline{V}^n$ again form a fundamental system of entourages.

Entourages $V$ such that $V = \overline{V}^1$ are called symmetric. If $V$ is any entourage, then $V \cap \overline{V}^1$ and $V \cup \overline{V}^1$ are symmetric entourages, and axioms (F_{II}) and (U_{II}) show that the symmetric entourages form a fundamental system of entourages.

A set $\mathfrak{B}$ of subsets of $X \times X$ is a fundamental system of entourages of a uniformity on $X$ if and only if $\mathfrak{B}$ satisfies axiom (B_I) of Chapter I, § 6, no. 3, and also satisfies the following axioms:

(U'_I) Every set of $\mathfrak{B}$ contains the diagonal $\Delta$ .

(U'_II) For each $V \in \mathfrak{B}$ there exists $V' \in \mathfrak{B}$ such that $V' \subset \overline{V}^1$ .

(U'_III) For each $V \in \mathfrak{B}$ there exists $W \in \mathfrak{B}$ such that $\overline{W}^2 \subset V$ .

If $X$ is not empty, a fundamental system of entourages of a uniform structure on $X$ is a base of the filter formed by the entourages of this structure (Chapter I, § 6, no. 3, Proposition 3).

(*) We recall (Set Theory, R, § 3, nos. 4 and 10) that if $V$ and $W$ are two subsets of $X \times X$ , then the set of pairs $(x, y) \in X \times X$ , such that $(x, z) \in W$ and $(z, y) \in V$ for some $z \in X$ , is denoted by $V \circ W$ or $VW$ , and that the set of pairs $(x, y) \in X \times X$ such that $(y, x) \in V$ is denoted by $\overline{V}^1$ .

Examples of uniformities. \* 1) On the set $\mathbf{R}$ of real numbers we can define a uniformity, called the additive uniformity, as follows: for each $\alpha > 0$ let $V_\alpha$ be the subset of $\mathbf{R} \times \mathbf{R}$ consisting of all pairs $(x, y)$ such that $|x - y| < \alpha$ ; as $\alpha$ runs through the set of all real numbers $> 0$ , the $V_\alpha$ form a fundamental system of entourages for the additive uniformity on $\mathbf{R}$ . Similarly we can define a uniformity (again called the additive uniformity) on the set $\mathbf{Q}$ of rational numbers; we shall study these structures, and analogous uniform structures on groups, in Chapters III and IV. \* 2) Let $X$ be a set and let $R$ be an equivalence relation on $X$ ; let $C$ be the graph of $R$ in $X \times X$ . Then $\Delta \subset C$ and $\overset{2}{C} = \overset{-1}{C} = C$ ; (Set Theory, R, § 5, no. 1); the set of subsets of $X \times X$ which consists of the set $C$ alone is therefore a fundamental system of entourages of a uniformity on $X$ . In particular, if we take $R$ to be the relation of equality, then $C = \Delta$ and the entourages of the corresponding uniformity are all the subsets of $X \times X$ which contain $\Delta$ ; this uniformity is called the discrete uniformity on $X$ , and the set $X$ endowed with this uniformity is called a discrete uniform space. 3) On the set $\mathbf{Z}$ of rational integers we can define a uniformity, important in the theory of numbers, as follows: given a prime number $p$ , let $W_n$ be the set of all pairs $(x, y) \in \mathbf{Z} \times \mathbf{Z}$ such that $x \equiv y \pmod{p^n}$ , for each integer $n > 0$ . It is easily verified that the sets $W_n$ (for a fixed $p$ ) form a fundamental system of entourages of a uniformity on $\mathbf{Z}$ , called the $p$ -adic uniformity (cf. Chapter III, § 6, Exercises 23 ff., and Chapter IX, § 3, no. 2).

In accordance with the general definitions (Set Theory, R, § 8, no. 5), if $X$ and $X'$ are two sets each endowed with uniformities whose sets of entourages are $U$ and $U'$ respectively, then a bijection $f$ of $X$ onto $X'$ is an isomorphism of the uniformity of $X$ onto that of $X'$ if $g(U) = U'$ , where $g = f \times f$ .

For example, if $X$ and $X'$ are two equipotent sets, then every bijection of $X$ onto $X'$ is an isomorphism of the discrete uniformity of $X$ onto the discrete uniformity of $X'$ .

2. Topology of a uniform space

Proposition 1

Let $X$ be a set endowed with a uniform structure $U$ , and for each $x \in X$ let $\mathfrak{B}(x)$ be the set of subsets $V(x)$ of $X (*)$ , where $V$ runs through the set of entourages of $U$ . Then there is a unique topology on $X$ such that, for each $x \in X$ , $\mathfrak{B}(x)$ is the neighbourhood filter of $x$ in this topology.

We have to show that the $\mathfrak{B}(x)$ satisfy conditions (V_I), (V_{II}), (V_{III}) and (V_{IV}) of Chapter I, § 1, no. 2. That this is so for the first three of

() We recall (Set Theory, R, § 3, no. 7*) that if $V$ is any subset of $X \times X$ and $x$ is any element of $X$ , then $V(x)$ denotes the set of all $y \in X$ such that $(x, y) \in V$ .

these conditions follows immediately from the fact that the entourages of $\mathfrak{U}$ satisfy $(\mathbf{F}_I),\ (\mathbf{F}_{II})$ and $(\mathbf{U}_I)$ . As to $(\mathbf{V}_{IV})$ , let $V$ be an entourage of $\mathfrak{U}$ , $W$ an entourage of $\mathfrak{U}$ such that $\dot{W} \subset V$ ; then if $(x, y) \in W$ and $(y, z) \in W$ we have $(x, z) \in V$ , so that $W(y) \subset V(x)$ for all $y \in W(x)$ , and therefore $V(x) \in \mathfrak{B}(y)$ for all $y \in W(x)$ . This completes the proof.

Definition 3

The topology defined in Proposition 1 is called the topology induced by the uniform structure $\mathfrak{U}$ .

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 uniformity on the set of rational numbers is the topology of the rational line.

Example 2

On any set $X$ , the topology induced by the discrete uniformity (no. 1, Example 2) is the discrete topology.

In the future, when we speak of the topology of a uniform space $X$ , we shall always mean the topology induced by the uniform structure of the space, unless the contrary is expressly stated. The topological space obtained by putting this topology on the set $X$ is sometimes called the topological space underlying the uniform space in question. For example, when we say that a uniform space is Hausdorff, or compact, or locally compact, etc., we mean that the underlying topological space has this property.

If $X$ and $X'$ are two uniform spaces, any isomorphism $f$ of the uniform structure of $X$ onto that of $X'$ is also a homeomorphism of $X$ onto $X'$ ; we say that $X$ is an isomorphism of the uniform space $X$ onto the uniform space $X'$ . It should be noted that a homeomorphism of $X$ onto $X'$ is not necessarily an isomorphism of the uniform structure of $X$ onto that of $X'$ .

In other words, distinct uniformities on the same set $X$ can induce the same topology. *For example, on ]0, +∞[, the additive and the multiplicative uniformities (which are distinct : Chapter III, § 6, Exercise 17) induce the same topology.* For another example see § 2, no. 2, Remark 1.

Proposition 2

Let $X$ be a uniform space. For every symmetric entourage $V$ of $X$ and every subset $M$ of $X \times X$ , $VMV$ is a neighbourhood of $M$ in the product space $X \times X$ , and the closure of $M$ in this space is given by the formula

\overline{M} = \bigcap_{V \in \mathscr{S}} VMV

where $\mathscr{S}$ denotes the set of symmetric entourages of $X$ .

Let $V$ be a symmetric entourage of $X$ . The relation $(x, y) \in VMV$ means that there is an element $(p, q)$ of $M$ such that $(x, p) \in V$ and $(q, y) \in V$ : in other words ( $V$ being symmetric) $x \in V(p)$ and $y \in V(q)$ , that is $(x, y) \in V(p) \times V(q)$ . Since $V(p) \times X(q)$ is a neighbourhood of $(p, q)$ in $X \times X$ , the first part of the proposition is proved. The relations $(x, p) \in V, (y, q) \in V$ can also be written $p \in V(x), q \in V(y)$ or $(p, q) \in V(x) \times V(y)$ . As $V$ runs through $\mathscr{S}$ , the sets $V(x) \times V(y)$ form a fundamental system of neighbourhoods of $(x, y)$ in $X \times X$ ; for if $U, U'$ are any two entourages there is always a symmetric entourage $V \subset U \cap U'$ , so that $V(x) \times V(y) \subset U(x) \times U'(y)$ . Hence $V(x) \times V(y)$ meets $M$ for each $V \in \mathscr{S}$ if and only if $(x, y) \in \overline{M}$ , and formula (1) follows.

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

\overline{A} = \bigcap_{V \in \mathscr{S}} V(A) = \bigcap_{U \in \mathfrak{U}} V(A)

where $\mathfrak{U}$ denotes the set of all entourages in $X$ .

If $M = A \times A$ , then $VMV = V(A) \times V(A)$ for any $V \in \mathscr{S}$ ; for the relation "there exists $p \in A$ such that $(x, p) \in V$ " is by definition equivalent to $x \in V(A)$ . The corollary now follows from Chapter I, § 4, no. 2, Proposition 5 and no. 3, Proposition 7.

$V(A)$ is said to be the V-neighbourhood of $A$ .

If $V$ is an entourage which is open in $X \times X$ , then $V(x)$ is open in $X$ for each $x \in X$ (Chapter I, § 4, no. 2, Corollary to Proposition 4) and therefore $V(A)$ , being the union of the $V(x)$ as $x$ runs through $A$ , is open in $X$ . On the other hand, if $V$ is a closed entourage in $X \times X$ , $V(A)$ need not be closed in $X$ for every subset $A$ of $X$ (Exercise 3).

It should also be remarked that as $V$ runs through the set of entourages of $X$ , the sets $V(A)$ do not necessarily form a fundamental system of neighbourhoods of $A$ in $X$ (Exercise 2).

Corollary 2

The interiors (resp. the closures) of the entourages of $X$ in $X \times X$ form a fundamental system of entourages of $X$ .

If $V$ is any entourage of $X$ , there is a symmetric entourage $W$ such that $W \subset V$ ; since $W$ is a neighbourhood of $W$ (Proposition 2), the interior of $V$ in $X \times X$ contains $W$ and is therefore an entourage. Furthermore, we have $W \subset W \subset W \subset V$ by Proposition 2, and hence $V$ contains the closure of an entourage.

Corollary 3

Every uniform space satisfies axiom* $(O_{III})$ .

If $x$ is any point of $X$ and $V$ runs through the entourages of $X$ which are closed in $X \times X$ , then the sets $V(x)$ form a fundamental system of neighbourhoods of $x$ in $X$ by Corollary 2, and they are closed in $X$ (Chapter I, , no. 2, Corollary to Proposition 4)

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 $\Delta$ of $X \times X$ . Every Hausdorff uniform space is regular.

The latter statement follows immediately from Corollary 3 to Proposition 2. We have seen that the closed entourages form a fundamental system of entourages (Proposition 2, Corollary 2); if their intersection is $\Delta$ , then $\Delta$ is closed in $X \times X$ and consequently $X$ is Hausdorff (Chapter I, § 8, no. 1, Proposition 1). Conversely, if $X$ is Hausdorff then for every point $(x, y) \notin \Delta$ there is an entourage $V$ of $X$ such that $y \notin V(x)$ , or equivalently $(x, y) \notin V$ ; hence $\Delta$ is the intersection of all the entourages.

If a uniform space $X$ is Hausdorff, we say that the uniform structure of $X$ is Hausdorff. If $\mathfrak{B}$ is a fundamental system of entourages for this structure; then $X$ is Hausdorff if and only if the intersection of all the sets of $\mathfrak{B}$ is $\Delta$ .

Exercises

See the exercises for § 1.