On dense subsets of Tychonoff products of \(T_{1}\)-spaces
type: Item
instance of: scholarly article
title: On dense subsets of Tychonoff products of \(T_{1}\)-spaces
zbMATH Open document ID: 1402.54003
published in: Topology and its Applications
publication date: 2018-10-02T00:00:00Z
review text: It is well-known that the Tychonoff product of \(\mathfrak c\) many separable spaces is a separable space (by the classical Hewitt-Marczewski-Pondiczery Theorem). In the paper under review, the author considers the problem of the existence, in products of \(\mathfrak c\) separable spaces, of a dense countable subset with additional properties. The main result states as follows: let \(Z\) be a separable, infinite topological space, and let \(\eta\) be a family of non-empty, countable subsets of \(Z\), \(|\eta| \leqslant \mathfrak c\). Then there is a countable dense subset \(Q \subseteq Z^{\mathfrak c}=\prod\limits_{X \subseteq \omega} ZX\) satisfying the following condition: (*) every countable set \(E \subseteq Q\) contains a countable subset \(E' \subseteq E\) such that for every \(G \in \eta\) there is some \(X \subseteq \omega\) such that \(\prod_X(E') = G\) and \(\prod_{X|_{E'}}: E' \to G\) is a finite-to-one mapping. The author deduces a number of consequences of the above stated result. For instance, it is shown in the paper that if \(Z\) is a separable \(T_1\) space with more than one point then there is a countable dense set \(Q \subseteq Z^{\mathfrak c}\) which does not contain nontrivial convergent sequences in \(Z^{\mathfrak c}\). All proofs of the paper use the notion of independent matrices of subsets of a countable set.
reviewed by: Samuel Gomes da Silva
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 6946127
zbMATH Keywords: product of spaces
zbMATH Keywords: dense set
zbMATH Keywords: convergent sequence
zbMATH Keywords: independent matrix
author: A. A. Gryzlov
MaRDI profile type: Publication
full work available at URL: j.topol.2018.09.003
OpenAlex ID: W2889640857
cites work: Q4150804
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Wikidata QID: Q129293934
DOI: 10.1016/J.TOPOL.2018.09.003
description: scientific article; zbMATH DE number 6946127