On Baire class one functions on a product space
type: Item
instance of: scholarly article
title: On Baire class one functions on a product space
zbMATH Open document ID: 1332.54156
author: V. V. Bykov
published in: Topology and its Applications
publication date: 2016-01-21T00:00:00Z
review text: Let \(X\) be a Tychonoff product of a countable collection of non-singleton completely regular spaces \(X_j\) and let \(S\) be a subset of \(X\). A function \(f:S\to\mathbb{R}\) is said to be finitely determined if there exists \(m\in\mathbb{N}\) such that \(f(x)=f(y)\) for all \(x,y\in S\) whose first \(m\) coordinates coincide. Let \(CF(S)\) be the set of all continuous finitely determined functions from \(S\) to \(\mathbb{R}\). The author shows that every continuous function \(f:X\to\mathbb{R}\) is the uniform limit of a sequence in \(CF(X)\) if and only if \(X\) is pseudocompact. Further, if \(X\) is metrizable, then every Baire class \(\alpha\) function \(f:S\to\mathbb{R}\) is the pointwise limit of a sequence of finitely determined functions in previous Baire classes if and only if \(\alpha=1\). Moreover, a lower semicontinuous function on a metrizable \(X\) is the pointwise limit of an increasing sequence in \(CF(X)\) if and only if all but finitely many of \(X_j\) are compact.
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 26A21
zbMATH DE Number: 6534246
zbMATH Keywords: Baire class
zbMATH Keywords: Tychonoff topology
zbMATH Keywords: semicontinuous function
reviewed by: Ján Borsík
MaRDI profile type: Publication
full work available at URL: j.topol.2015.12.003
OpenAlex ID: W2217766679
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DOI: 10.1016/J.TOPOL.2015.12.003
description: scientific article; zbMATH DE number 6534246