Position of subspaces in spaces: Relative versions of compactness, Lindelöf properties, and separation axioms
type: Item
instance of: scholarly article
title: Position of subspaces in spaces: Relative versions of compactness, Lindelöf properties, and separation axioms
zbMATH Open document ID: 0745.54003
author: Kh. M. M. Genedi
author: A. V. Arhangel'skii
published in: Moscow University Mathematics Bulletin
publication date: 1992-06-25T00:00:00Z
review text: The authors define relative versions of certain standard topological properties and state certain theorems and examples concerning them. No proofs are given. Let \(X\) be a topological space and let \(Y\) be a subspace of \(X\). The authors say that \(Y\) is regular (superregular) in \(X\) if for each \(y\in Y\) and any set \(A\) which is closed in \(X\) and not containing \(y\) there are disjoint subsets \(U\) and \(V\) open in \(X\) such that \(y\in U\) and \(A\cap Y\subseteq V\) (respectively, \(y\in U\) and \(A\subseteq V\)). They define \(Y\) to be strongly regular in \(X\) if for each \(x\in X\) and each set \(F\) not containing \(x\) and closed in \(X\) there are disjoint open sets \(U\), \(W\) in \(X\) such that \(x\in U\) and \(F\cap Y\subseteq W\). Proposition 1: There exists a \(T_ 2\)-space \(X\) having a countable closed discrete subspace \(Y\) such that \(Y\) is not regular in \(X\). A space \(Y\) is called strongly normal in a space \(X\) if for any two disjoint sets \(A\) and \(B\) closed in \(Y\) there are disjoint open sets \(U\) and \(V\) in \(X\) such that \(A\subseteq U\) and \(B\subseteq V\). Theorem 1: Suppose \(Y\) is regular in \(X\) and \(Y\) is a Lindelöf subspace. Then \(Y\) is strongly normal in \(X\). Propositions 6 and 7: If \(Y\) is compact in a \(T_ 2\)-space \(X\), then \(Y\) is strongly regular in \(X\), but \(Y\) is not necessarily superregular in \(X\). The paper contains other generalizations of classical theorems of general topology and several examples showing the scope and limitations of the results obtained. The paper concludes with a problem.
Mathematics Subject Classification ID: 54B05
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 7030
zbMATH Keywords: supernormal subspace
zbMATH Keywords: superregular subspace
reviewed by: Howard Wicke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7030