Points of continuity of quasi-continuous functions
type: Item
instance of: scholarly article
title: Points of continuity of quasi-continuous functions
zbMATH Open document ID: 1479.54030
author: Marzieh Neghaban
author: Alireza Kamel Mirmostafaee
published in: Tatra Mountains Mathematical Publications
publication date: 2021-12-06T00:00:00Z
review text: Fix a function \(f:X\times Y\to Z\) for topological spaces \(X\), \(Y\) and \(Z\). Then \(f\) is KC if it is quasi-continuous with respect to the first variable and continuous with respect to the second, and is Weston if \(\{x\in X\mid (x,y)\in C(f)\}\) is residual for each \(y\in Y\), where \(C(f)\) is the set of points of continuity. The space \(Y\) is Weston with respect to \(Z\) provided for all Baire \(X\) and all \(f\) if \(f\) is KC it is also Weston. Then \(Y\) is Weston with respect to every weakly developable completely regular space if and only if it is Weston with respect to \(\mathbb R\). Several conditions are given under which we can deduce that \(f\) has the Weston property, for example if \(X=\prod_{i=1}^nX_i\) where \(X_1\) is Baire, each \(X_i \ (i>1)\) has a countable \(\pi\)-base, the product of each \(X_i \ (i>1)\) with a Baire space is Baire, \(Y\) is Weston with respect to \(Z\), \(Z\) is regular and \(f\) is quasi-continuous with respect to each \(X_i\) and continuous with respect to \(Y\). Making use of a game introduced by \textit{G. Gruenhage} [General Topology Appl. 6, 339--352 (1976; Zbl 0327.54019)], conditions are given under which the set of points of continuity of a function forms a (dense) \(G_\delta\) subset of the domain.
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 54E52
zbMATH DE Number: 7439076
zbMATH Keywords: joint continuity
zbMATH Keywords: separate continuity
zbMATH Keywords: quasi-continuity
zbMATH Keywords: closed graph
reviewed by: David B. Gauld
MaRDI profile type: Publication
full work available at URL: tmmp-2020-0017
OpenAlex ID: W3096780539
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DOI: 10.2478/TMMP-2020-0017
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description: scientific article; zbMATH DE number 7439076