Zero-dimensional proximities and zero-dimensional compactifications
type: Item
instance of: scholarly article
title: Zero-dimensional proximities and zero-dimensional compactifications
zbMATH Open document ID: 1165.54010
author: Guram Bezhanishvili
published in: Topology and its Applications
publication date: 2009-04-29T00:00:00Z
review text: It is shown that the poset of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space \(X\) is isomorphic to the poset of zero-dimensional proximities on \(X\) that induce the topology on \(X\). That poset is also isomorphic to the poset of Boolean bases of \(X\). The author also states that for a regular extremally disconnected space \(X\), the Stone-Čech compactification of \(X\) is a unique (up to equivalence) extremally disconnected compactification of \(X\). We note that related investigations about transitive totally bounded quasi-uniformities were conducted in asymmetric topology, see e.g., the article of \textit{A. Losonczi} [Quaest. Math. 23, No.~1, 67--75 (2000; Zbl 0958.54033)] where lattice-bases were studied or that of \textit{H. P. A. Künzi} and \textit{G. C. L. Brümmer} [Math. Proc. Camb. Philos. Soc. 101, 237--247 (1986; Zbl 0618.54024)] where those compact \(T_2\)-ordered spaces whose determining (totally bounded) quasi-uniformity is transitive were characterized.
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54G05
zbMATH DE Number: 5549312
zbMATH Keywords: compactification
zbMATH Keywords: proximity
zbMATH Keywords: zero-dimensional space
zbMATH Keywords: extremally disconnected space
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.036
OpenAlex ID: W2134863655
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DOI: 10.1016/J.TOPOL.2008.12.036
description: scientific article; zbMATH DE number 5549312