Countable compacta of continuity for projective functions
type: Item
instance of: scholarly article
title: Countable compacta of continuity for projective functions
zbMATH Open document ID: 1223.54049
author: Václav Vlasák
published in: Topology and its Applications
publication date: 2011-05-16T00:00:00Z
review text: For a Polish space \(X\) let \(\mathcal{K}(X)\) denote the space of all compact subsets of \(X\) equipped with the Vietoris topology. In the paper under review the author investigates connections between complexity of a function \(f\) from a Polish space \(X\) to a Polish space \(Y\) and complexity of the set \(\widetilde C(f)=\{K\in \mathcal{K}(X)\); \(f|_K\) is continuous and \(K\) has exactly one limit point\}. In the main result he shows that for each positive integer \(n\) the following conditions are equivalent: (1)~\(\widetilde{C}(f)\in\Delta_n^1(\mathcal{K}(X))\); (2)~\(\widetilde{C}(f)\in \Sigma^1_n(\mathcal{K}(X))\); (3)~\(f\) is \(\Delta_n^1\)-measurable. In particular, \(f\) is Borel iff \(\widetilde{C}(f)\) is Borel.
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 5888211
zbMATH Keywords: continuity
zbMATH Keywords: Vietoris topology
zbMATH Keywords: Borel sets
zbMATH Keywords: projective classes
zbMATH Keywords: determinacy axiom.
MaRDI profile type: Publication
full work available at URL: j.topol.2011.03.005
OpenAlex ID: W2013653182
cites work: Ideals of compact sets associated with Borel functions.
cites work: Collections of compact sets and functions having \(G_\delta\)-graphs
cites work: Q4323294
cites work: The Structure of σ-Ideals of Compact Sets
cites work: Compact sets of continuity for Borel functions
DOI: 10.1016/J.TOPOL.2011.03.005
description: scientific article; zbMATH DE number 5888211