Some properties of subcompact spaces
type: Item
instance of: scholarly article
title: Some properties of subcompact spaces
zbMATH Open document ID: 1494.54016
DOI: 10.1134/S0001434622010229
author: V. I. Belugin
author: Alexander V. Osipov
author: E. G. Pytkeev
published in: Mathematical Notes
publication date: 2022-03-14T00:00:00Z
review text: Readers must guess that ZFC is the set-theoretic framework of this article. A continuous bijection of a topological space \(X\) onto a topological space \(Y\) is called a condensation. A compact Hausdorff space \(X\) is called: (1) an \(a\)-space if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a compact Hausdorff space; (2) a strict \(a\)-space if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a compact Hausdorff space \(Y\) which can be extended to a continuous mapping of \(X\) onto \(Y\). A Hausdorff space \(X\) is called: (3) subcompact if there exists a condensation of \(X\) onto a compact Hausdorff space; (4) \(a\)-subcompact if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto an \(a\)-space; (5) strictly \(a\)-subcompact if, for each countable subset \(C\) of \(X\), there exists a condensation of \(X\setminus C\) onto a strict \(a\)-space \(Y\) which has an extension to a continuous mapping of \(X\) onto \(Y\); (6) weakly \(a\)-subcompact if, for every countably infinite subset \(A\) of \(X\), there exists a countable subset \(B\) of \(X\) such that \(A\subseteq B\) and there exists a condensation of \(X\setminus B\) onto a compact Hausdorff space. A compact weakly \(a\)-subcompact space is called a weak \(a\)-space. A Hausdorff space is dyadic if it is a continuous image of a Cantor cube. The authors generalize their result from [\textit{V. I. Belugin} et al., Math. Notes 109, No. 6, 849--858 (2021; Zbl 1473.54010)] by proving that, for every compact Hausdorff space \(X\) and every dense-in-itself dyadic Hausdorff space \(Y\), the Cartesian product \(X\times Y\) is a strict \(a\)-space. Furthermore, among other facts, the authors show what follows. If a Hausdorff dyadic space \(Y\) has an isolated point, and \(X\) is a compact Hausdorff space which is not a strict \(a\)-space, then the product \(X\times Y\) is not …
reviewed by: Eliza Wajch
Mathematics Subject Classification ID: 54C05
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54E45
zbMATH DE Number: 7488494
zbMATH Keywords: continuous bijection
zbMATH Keywords: condensation
zbMATH Keywords: \textit{a}-space
zbMATH Keywords: strict \textit{a}-space
zbMATH Keywords: weak \textit{a}-space
zbMATH Keywords: dyadic Hausdorff space
zbMATH Keywords: subcompact space
MaRDI profile type: Publication
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full work available at URL: s0001434622010229
OpenAlex ID: W4213436606
description: scientific article; zbMATH DE number 7488494