Arkhangel’skij, A. V.; Tartir, J. A characterization of compactness by a relative separation property. (English) Zbl 0851.54001 Quest. Answers Gen. Topology 14, No. 1, 49-52 (1996). The authors prove a surprising and quite interesting characterization of compactness by using so-called relative separation axioms. A subspace \(Y\) of a topological space \(X\) is said to be regular in \(X\) if for each point \(y\) of \(Y\) and every subset \(P\) of \(X\) which is closed in \(X\) and does not contain \(y\) there exist disjoint (in \(X\)) open sets \(U\) and \(V\) such that \(U\) contains \(y\) and \(P\cap Y\) is contained in \(V\). The authors show that a Hausdorff space \(Y\) is compact if and only if it is regular in every larger Hausdorff space \(X\). In addition, some related questions and examples are discussed, in particular they discuss the question whether a regular space \(Y\) which is strongly normal in every larger regular space \(X\) has to be Lindelöf. Reviewer: M.Ganster (Graz) Cited in 1 ReviewCited in 4 Documents MSC: 54A05 Topological spaces and generalizations (closure spaces, etc.) 54D30 Compactness 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) Keywords:relative regularity; relative separation axioms; strongly normal PDFBibTeX XMLCite \textit{A. V. Arkhangel'skij} and \textit{J. Tartir}, Quest. Answers Gen. Topology 14, No. 1, 49--52 (1996; Zbl 0851.54001)