Ideals of compact sets associated with Borel functions.
type: Item
instance of: scholarly article
title: Ideals of compact sets associated with Borel functions.
zbMATH Open document ID: 1053.54042
author: Francis Jordan
published in: Real Analysis Exchange
publication date: 2004-08-06T00:00:00Z
review text: For a Polish space \(X\), let \(J (X)\) be the hyperspace of compact subsets of \(X\) with the Hausdorff metric. Given a function \(f: X\to Y\) between Polish spaces \(X\) and \(Y\), put \({\mathcal C} (f) = \{ C \in J(X) : f \upharpoonright C\) is continuous\(\}\). The author establishes a number of results about the connection between the complexity of \(f\) and the complexity of \({\mathcal C} (f)\). For example, \(f\) is continuous iff \({\mathcal C} (f)\) is a \(G_\delta\)-subset of \(J(X)\). Also, if \({\mathcal C} (f)\) is a \(\Pi^0_3\) subset of \(J(X)\), then \(f\) is a Baire class one function which in turn implies that \({\mathcal C} (f)\) is a \(\Pi^0_4\) subset of \(J(X)\), with none of the two implications being reversible. Furthermore, for a Borel function \(f\), \(f\) has a \(G_\delta\)-graph iff \({\mathcal C} (f)\) is a Borel subset of \(J(X)\). The proofs use arguments from classical descriptive set theory.
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A21
zbMATH DE Number: 2084016
zbMATH Keywords: descriptive set theory
zbMATH Keywords: Polish space
zbMATH Keywords: Borel sets
zbMATH Keywords: hyperspace of compact sets
zbMATH Keywords: Hausdorff metric
zbMATH Keywords: Baire class of functions
MaRDI profile type: Publication
OpenAlex ID: W1490929562
DOI: 10.14321/REALANALEXCH.28.1.0015
Recommended article: Q3746048
Recommended article: Q4445274
Recommended article: On Borel ideals
Recommended article: Some properties of Borel ideals
Recommended article: Compact sets of continuity for Borel functions
Recommended article: Invariant Ideals and Borel Sets
Recommended article: Compactness with ideals
Recommended article: Cofinalities of Borel ideals
Recommended article: Ideals of functions which achieve zero on a compact set
Recommended article: Borel sets and functions in topological spaces
description: scientific article; zbMATH DE number 2084016