Typical properties of continuous functions via the Vietoris topology
type: Item
instance of: scholarly article
title: Typical properties of continuous functions via the Vietoris topology
zbMATH Open document ID: 0828.54014
author: M. Balcerzak
published in: Real Analysis Exchange
publication date: 1996-01-10T00:00:00Z
review text: Let \(X\) be a compact metrizable space and let \(C(X)\) be the Banach space of all continuous real functions. Let \(K(X)\) be the topological space (with the Vietoris topology) of all nonempty closed sets in \(X\). The main result is the following: Let \(X\) be a closed subset of \(I = [0,1]\) and let \(A\) be the family of all analytic functions on \(I\). If \(E \subset K(X)\) is a \(G^*_\delta\) set containing all finite sets, then for any set \(D \subset A\) such that \(A \smallsetminus D\) is dense in \(C(I)\), the set of all \(f \in D \smallsetminus X\) such that for all \(h \in D\) the set \(\{x; f = h\} \in E\) is dense of type \(G_\delta\).
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 54E50
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 465114
zbMATH Keywords: Banach space continuous functions
zbMATH Keywords: Vietoris topology
zbMATH Keywords: typical properties
zbMATH Keywords: residual set
zbMATH Keywords: analytic functions
reviewed by: Zbigniew Grande
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 465114