More on separation of a diagonal
type: Item
instance of: scholarly article
title: More on separation of a diagonal
zbMATH Open document ID: 1203.54016
author: Dennis Burke
author: Raushan Z. Buzyakova
published in: Topology and its Applications
publication date: 2010-08-27T00:00:00Z
review text: Following the second author a completely regular \(T_1\)-space \(X\) is called functionally \(\Delta\)-paracompact if for every \(A\subset X^2\setminus\Delta_X\) closed in \(X^2\) there exists a locally finite cover \({\mathcal U}\) of \(X\) consisting of co-zero sets such that \(\bigcup\{U\times U\mid U\in{\mathcal U}\}\) does not meet \(A\). It is known from the second author's paper [ibid. 157, No.~2, 352--358 (2010; Zbl 1187.54020)] that every paracompact Hausdorff-space and every GO-space is \(\Delta\)-paracompact. It is also known that a completely regular \(T_1\)-space is \(\Delta\)-paracompact if and only if it is divisible in the sense of \textit{Á. Császár} [General topology. Bristol: Adam Hilger Ltd. Budapest: Akademiai Kiado. 488 p. (1978; Zbl 0366.54001)]. In this paper the following is shown. Theorem 1. Every first countable, countably compact, monotonically normal space is functionally \(\Delta\)-paracompact. A completely regular \(T_1\)-space \(X\) is called regular \(\Delta\)-paracompact if for every \(A\subset X^2\setminus\Delta_X\) closed in \(X^2\) there exists a locally finite open cover \({\mathcal U}\) of \(X\) such that \(\bigcup\{\text{cl\,} U\times\text{cl\,} U\mid U\in{\mathcal U}\}\) does not meet \(A\). Theorem 2. Every regular \(\Delta\)-paracompact space is collectionwise normal. It is known that functionally \(\Delta\)-paracompactness implies regular \(\Delta\)-paracompactness. Question 1. Is every regular \(\Delta\)-paracompact space functionally \(\Delta\)-paracompact? Obviously, every regular \(\Delta\)-paracompact space is \(\Delta\)-paracompact, i.e., has the property that for every \(A\subset X^2\setminus\Delta_X\) closed in \(X^2\) there exists a locally finite open cover \({\mathcal U}\) of \(X\) such that \(\bigcup\{U\times U\mid U\in{\mathcal U}\}\) does not meet \(A\). Theorem 3. Every \(\Delta\)-paracompact space is strongly collectionwise Hausdorff. Theorem 4. Every \(\Delta\)-paracompact pseudocompact space is countably …
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 5775742
zbMATH Keywords: functionally \(\Delta\)-paracompact space
zbMATH Keywords: monotonically normal space
zbMATH Keywords: regular \(\Delta\)-paracompact space
zbMATH Keywords: collectionwise normal space
zbMATH Keywords: \(\Delta\)-paracompact space
zbMATH Keywords: strongly collectionwise Hausdorff space
zbMATH Keywords: functionally \(\Delta\)-normal space
zbMATH Keywords: \(\Delta\)-normal space
zbMATH Keywords: normal space
zbMATH Keywords: \(G_\delta\)-diagonal
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
full work available at URL: j.topol.2010.06.006
OpenAlex ID: W1967053126
cites work: Normality and Martin's axiom
cites work: Monotone normality
cites work: Metrization of Topological Spaces
cites work: Q4769645
cites work: Separation of a diagonal
cites work: Q4142325
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cites work: Spaces for which the diagonal has a closed neighborhood base
cites work: Screenability, Pointwise Paracompactness, and Metrization of Moore Spaces
cites work: On normality and countable paracompactness
cites work: Spaces with regular \(G_ \delta\)-diagonals
DOI: 10.1016/J.TOPOL.2010.06.006
description: scientific article; zbMATH DE number 5775742