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IX. Utilisation des nombres réels en topologie générale
App. 1. Espaces de Lindelöf
- Definition 1 One says that a topological space X (not necessarily separated) is a Lindelöf space if, from every open covering of X, one can extract a countable covering of X. 0ED4
- Proposition 1 Let X be a topological space. 0ED5
- Corollary Every sub-Lindelöf space, and in particular every Polish or Lusin space, is a Lindelöf space. 0ED6
- Proposition 2 Every regular Lindelöf space is paracompact. 0ED7
- Corollary Every regular sublinian space is paracompact (and in particular normal (IX, p. 49, prop. 4) and a fortiori completely regular). 0ED8
- Proposition 3 Let X be a topological space (not necessarily separated). The following properties are equivalent: a) Every open subspace of X is a Lindelöf space. b) Every set of lower (resp. upp… 0ED9
- Corollary 1 Let X be a regular souslinian space, H a set of continuous numerical functions in X, which separates the points of X (IX, p. 9, def. 5). There then exists a countable subset H' of… 0EDA
- Corollary 2 Every compact souslinian space is metrizable. 0EDB
App. 2. Produits infinis dans les algèbres normées
- Definition 1 One says that the sequence (xₙ)_n ≥ N is multipliable in the normed algebra A if the mapping J ↦ p_J has a limit following the filter of sections of the set F(N) of finite subsets… 0EDC
- Remark 1 When A is a commutative algebra, def. 1 is identical with that which was given in III, p. 37 (Remark 3); but when A is not commutative, the order structure of the set of indices N… 0EDD
- Remark 2 Def. 1 generalizes immediately to the case of a family (xₙ)_n ∈ I whose set of indices I is a subset of Z (totally ordered by the order induced by that of Z); we leave to the reade… 0EDE
- Theorem 1 Let (xₙ)_n ∈ N be a sequence of points of a complete normed algebra A. 0EDF
- Remark When (xₙ) is multipliable and has a noninvertible product, the condition of th. 1 is no longer necessarily verified; for example, if all the xₙ are equal to the same element x such… 0EDG
- Corollary 1 If (xₙ) is a multipliable sequence whose product is invertible in A, lim_n → ∞ xₙ = e. 0EDH
- Corollary 2 If (xₙ) is a multipliable sequence whose product is invertible in A, any sequence (x_nₖ)_k ∈ N extracted from (xₙ) ((nₖ) being a strictly increasing sequence of integers) is multip… 0EDI
- Theorem 2 Let A be a complete normed algebra; if (uₙ) is an absolutely convergent series of points of A, the sequence (e + uₙ) is multipliable in A; moreover, if all the elements e + uₙ are… 0EDJ
- Corollary If the series of general term uₙ is absolutely convergent, and if none of the elements e + uₙ is a divisor of 0 in A, the product ∏_n ∈ N (e + uₙ) is not a divisor of 0 in A. 0EDK
- 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 invertible, the series of general term uₙ is absolutely convergent. 0EDL
- Lemma 1 For every finite subset I of N, let φ(I) = sup_J ⊂ I |p_J - e|. For every subset J of I, one has |p_J - e - s_J| ≤ φ(I) σ_J. The lemma is evident if J is empty; let us prove it by… 0EDM
- Lemma 2 If I is a finite subset of N such that φ(I) < 1/c, one has σ_I ≤ (c φ(I))/(1 - c φ(I)). 0EDN
- Proposition 2 Let (xₙ) be a sequence of points of a complete normed algebra A. 0EDO
- Corollary 1 If the infinite product of general factor xₙ is convergent, and if ∏ₙ₌₀^∞ xₙ is invertible, lim_n → ∞ xₙ = e. 0EDP
- Corollary 2 If the infinite product of general factor xₙ is convergent, and if ∏ₙ₌₀^∞ xₙ is invertible, the infinite product of general factor yₙ = xₙ₊ₕ (n ≥ 0) is convergent. 0EDQ
- Proposition 3 Let (kₙ) be a strictly increasing sequence of integers geqslant 0 (k₀ = 0) : if the infinite product of general factor xₙ converges, and if one sets uₙ = ∏_p=kₙ^kₙ₊₁-1 xₙ, the infi… 0EDR