Functional countability is preserved by some products
type: Item
instance of: scholarly article
title: Functional countability is preserved by some products
published in: Acta Mathematica Hungarica
publication date: 2022-11-09T00:00:00Z
review text: In this paper all spaces are Tychonoff. A space \(X\) is functionally countable if every real continuous image of \(X\) is countable. \(X\) is exponentially separable if for every family \(\mathcal{F}\) of closed subsets of \(X\), there is a countable subset of \(X\) which meets every non-empty intersection of members of \(\mathcal{F}\). From the author's paper [Commentat. Math. Univ. Carol. 59, No. 3, 399--409 (2018; Zbl 1424.54071)] an exponentially separable space is functionally countable, a scattered Lindelöf space is exponentially separable and a compact functionally countable space is scattered. In the paper under review, the author examines the behaviour of functional countability under products. The main results are as follows: If \(X\) is functionally countable and \(K\) is a scattered Corson compact space, then \(X \times K\) is functionally countable. If a functionally countable space \(X\) is weakly Lindelöf or separable or has the Souslin property, then \(X \times L\) is functionally countable for any scattered Lindelöf space \(L\). A pseudocompact product of finitely many functionally countable spaces is functionally countable. The product of finitely many functionally countable, countably compact spaces is functionally countable. The product of two exponentially separable, countably compact spaces is exponentially separable provided one of them is sequential.\par The paper concludes with a list of ten open problems.
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54G12
Mathematics Subject Classification ID: 54G10
Mathematics Subject Classification ID: 54D65
zbMATH DE Number: 7614064
zbMATH Keywords: functionally countable space
zbMATH Keywords: compact space
zbMATH Keywords: \(P\)-space
zbMATH Keywords: scattered space
zbMATH Keywords: Corson compact space
zbMATH Keywords: product
zbMATH Keywords: sequential space
zbMATH Keywords: countably compact space
zbMATH Keywords: weakly Lindelöf space
author: Vladimir Tkachuk
reviewed by: Michael G. Charalambous
MaRDI profile type: Publication
full work available at URL: s10474-022-01258-7
OpenAlex ID: W4294954964
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cites work: Q3675172
DOI: 10.1007/S10474-022-01258-7
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description: scientific article; zbMATH DE number 7614064