Mutually compactificable topological spaces
type: Item
instance of: scholarly article
title: Mutually compactificable topological spaces
zbMATH Open document ID: 1144.54014
DOI: 10.1155/2007/70671
author: Martin Maria Kovár
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/54471
review text: No separation axioms are assumed in topological spaces in this paper, which first discusses the notion of \(\theta\)-regularity of a topological space \(X\), i.e., every filter base of \(X\) with a \(\theta\)-cluster point has a cluster point. (A point \(x\in X\) is \(\theta\)-cluster point if \(x \in \cap \{\text{cl}_{\theta}F \mid F \in\) the filterbase\}, where \(x\) is in the \(\theta\)-closure of \(F\), \(\text{cl}_{\theta}F\), if every closed neighborhood of \(x\) intersects \(F\).) A space \(X\) is said to be compactificable by a space \(Y\) (or \(X\) and \(Y\) are mutually compactificable) if \(X \cap Y = \emptyset\) and there exists a compact topology on \(K = X \cup Y\) extending the topologies on \(X\) and \(Y\) such that two points in \(X\) and in \(Y\), respectively, are separated in \(K\), i.e., they have disjoint neighborhoods. A \(\theta\)-regular space is compactificable by its Wallman remainder, \(\omega X \setminus X\), which is also \(\theta\)-regular. The author shows that a necessary and sufficient condition for \(X\) to be compactificable by a space Y is that X is \(\theta\)-regular. \textit{A. Császár}'s separation axiom \(S_2\) [Introduction to general topology. Translated by Mrs. A. Csaszar. Bristol: Adam Hilger Ltd. Budapest: Akademiai Kiado (1978; Zbl 0366.54001)] is used: Two points with an open set containing only one of them necessarily have open disjoint neighborhoods. A space is regular (no \(T_1\) assumed) if and only if it is \(\theta\)-regular and \(S_2\). A regular space on which every continuous real-valued function is constant (hence not \(T_{3.5}\)) is mutually compactificable by no \(S_2\) space. An example of a regular non-\(T_{3.5}\) space is shown that is mutually compactificable with the infinite countable discrete space. Regular spaces do not form a closed class with respect to mutual compactificability. Any two disjoint strongly locally compact (i.e., every point in the space has a compact closed neighborhood) spaces ar…
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D40
zbMATH DE Number: 5282467
zbMATH Keywords: Wallman compactification
zbMATH Keywords: mutual compactificable
zbMATH Keywords: theta-regular
zbMATH Keywords: regular
zbMATH Keywords: strongly locally compact
zbMATH Keywords: locally compact
reviewed by: Pao-sheng Hsu
MaRDI profile type: Publication
OpenAlex ID: W1968672091
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description: scientific article; zbMATH DE number 5282467