A selection theorem for \(C\)-spaces
type: Item
instance of: scholarly article
title: A selection theorem for \(C\)-spaces
zbMATH Open document ID: 0921.54014
DOI: 10.1016/S0166-8641(97)00159-4
published in: Topology and its Applications
publication date: 1999-09-29T00:00:00Z
review text: Un espace topologique \(X\) a la propriété \(C\) si, pour toute suite \(\{U_n \}_{n=1}^\infty\) de recouvrements ouverts de \(X\), il existe une suite \(\{{\mathcal V}_n \}_{n=1}^\infty\) de familles d'ouverts disjoints de \(X\) telle que \(\bigcup_{n=1}^\infty {\mathcal V}_n\) recouvre \(X\) et que \({\mathcal V}_n\) soit plus fine que \({\mathcal U}_n\) pour tout \(n\). L'auteur étudie ici l'existence de sélections continues de fonctions multivoques définies sur un espace ayant la propriété \(C\). En particulier, il prouve que, pour un espace paracompact \(X\), les conditions suivantes sont équivalentes: (1) \(X\) a la propriété \(C\); (2) Pour tout espace topologique \(Y\) et tout ouvert \(V\) de \(X\times Y\) tel que, pour tout \(x\in X\), l'ensemble \(V_x= \{y\in Y: (x,y)\in V\}\) soit non vide et contractile, la fonction \(x\to V_n\) a une sélection continue; (3)=(2) avec la restriction que \(Y\) est supposé être un espace de Banach; (4) Si \(\Phi\) est une fonction associant à tout point \(x\) de \(X\) un sous-ensemble \(\Phi(x)\) d'un espace topologique \(Y\) et vérifiant (i) pour tout \(x\in X\), \(\Phi(x)\) a le type d'homotopie faible d'un point, (ii) pour tout compact \(K\) de \(Y\), \(\{x\in K: K\subset \Phi(x)\}\) est ouvert dans \(X\), alors \(\Phi\) a une sélection continue.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 55S40
zbMATH DE Number: 1307996
author: Vladimir V. Uspenskij
reviewed by: Robert Cauty
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307996