Relative topological properties and relative topological spaces
type: Item
instance of: scholarly article
title: Relative topological properties and relative topological spaces
zbMATH Open document ID: 0848.54016
DOI: 10.1016/0166-8641(95)00086-0
published in: Topology and its Applications
publication date: 1996-09-04T00:00:00Z
review text: This is a highly interesting survey article on so-called relative topological properties. It has to do with the question of how a given subspace \(Y\) of a space \(X\) is ``located'' in \(X\). The basic idea is as follows: with each topological property \(\mathbb{P}\) one can associate a relative version of it formulated in terms of location of \(Y\) in \(X\) in such a way that when \(Y\) coincides with \(X\) then this relative property coincides with \(\mathbb{P}\). In the first part of this paper relative separation axioms are considered. Most results revolve around the situation when a subspace \(Y\) is normal (resp. strongly normal) in \(X\). \(Y \subseteq X\) is normal in \(X\) if for each pair \(A,B\) of closed disjoint subsets of \(X\) there are disjoint open subsets \(U\) and \(V\) in \(X\) such that \(A \cap Y \subseteq U\) and \(B \cap Y \subseteq V\). \(Y \subseteq X\) is strongly normal in \(Y\) if for each pair \(A,B\) of closed in \(Y\) disjoint subsets of \(Y\) there are open disjoint subsets \(U\) and \(V\) in \(X\) such that \(A \subseteq U\) and \(B \subseteq V\). In the second part of the paper it is discussed how several relative compactness type properties influence relative separation properties. Typical results mentioned there are: (i) If \(Y \) is Lindelöf in a regular space \(X\) then \(Y\) is paracompact in \(X\), and (ii) if \(Y\) is paracompact in \(X\), and \(X\) is regular, then \(Y\) is normal in \(X\). Throughout the paper a large number of still open problems are posed, thus setting the direction for possible future research.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 896606
zbMATH Keywords: relative property
zbMATH Keywords: relative compactness
zbMATH Keywords: relative separation properties
Wikidata QID: Q56535312
author: A. V. Arhangel'skii
reviewed by: Maximilian Ganster
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00086-0
OpenAlex ID: W1981735727
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description: scientific article; zbMATH DE number 896606