\(G_ \delta\)-open functionally bounded subsets in topological groups
type: Item
instance of: scholarly article
title: \(G_ \delta\)-open functionally bounded subsets in topological groups
zbMATH Open document ID: 0795.22002
DOI: 10.1016/0166-8641(93)90122-T
author: Salvador Hernández
author: Manuel Sanchis
published in: Topology and its Applications
publication date: 1994-03-10T00:00:00Z
review text: For a topological space \(X\), let \(C(X)\) denote the ring of all continuous real-valued functions on \(X\). A subset \(A\) of \(X\) is called functionally bounded (or relatively pseudocompact) in \(X\) if each \(f\in C(X)\) is bounded on \(A\), and \(A\) is said to be hyperbounded in \(X\) when \(\inf\{f(x)\): \(x\in A\}>0\) for every \(f\in C(X)\) which is positive on \(A\) [cf. \textit{T. Isiwata}, Pac. J. Math. 20, 455-480 (1967; Zbl 0149.405)]. A subset \(A\) of \(X\) is said to be \(G_ \delta\)-open in \(X\) when for any point \(x\in A\), there is a \(G_ \delta\)-set \(U\) in \(X\) such that \(x\in U\subset A\), and call \(A\) \(G_ \delta\)-dense in \(X\) if every \(G_ \delta\)-set in \(X\) intersects \(A\). This paper deals with functionally bounded sets of a topological group which are \(G_ \delta\)-dense in their closure in the bilateral completion \({\mathcal G}\) of the group. The following main results are proved: Theorem 1. Suppose that \(A\) is a functionally bounded subset of a topological group \(G\). Then \(A\) is hyperbounded if and only if \(A\) is \(G_ \delta\)-dense in \(cl_{\mathcal G} A\). -- Theorem 2. If \(A\) is a functionally bounded \(G_ \delta\)-open subset of a topological group \(G\), then the Cartesian product of \(cl_ G A\) by any pseudocompact space is a pseudocompact space. -- Theorem 3. Let \(G\) be a topological group and let \(B\) be a functionally bounded set in \(G\) which contains a \(G_ \delta\)-open subset \(A\) in \(B\). Then \(B\) is hyperbounded \(\Leftrightarrow B\) is pseudocompact \(\Leftrightarrow B\) is \(C\)-embedded in \(G\Leftrightarrow B\) is \(C\)-embedded in \(cl_{\mathcal G} B\Leftrightarrow B\) is \(P\)-embedded in \(G\Leftrightarrow B\) is \(P\)-embedded in \(cl_{\mathcal G} B\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54C50
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 500679
zbMATH Keywords: functionally bounded sets
zbMATH Keywords: topological group
zbMATH Keywords: completion
zbMATH Keywords: pseudocompact space
MaRDI profile type: Publication
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full work available at URL: 0166-8641(93)90122-t
OpenAlex ID: W1982522022
description: scientific article; zbMATH DE number 500679