Almost compact subspaces of hyperextensions
type: Item
instance of: scholarly article
title: Almost compact subspaces of hyperextensions
zbMATH Open document ID: 0831.54021
DOI: 10.1007/BF01874468
author: Ákos Császár
published in: Acta Mathematica Hungarica
publication date: 1996-02-13T00:00:00Z
review text: Given a space \(X\) together with an open subbase \(S\) for its topology, the author earlier constructed and studied a superspace \(h(X,S)\) [Period. Math. Hung. 1, 55-80 (1971; Zbl 0219.54021)]. His construction can be considered a generalization of the one used by \textit{J. de Groot} to obtain superextensions [Contrib. Extens. Theory topol. Struct., Proc. Sympos. Berlin 1967, 89-90 (1969; Zbl 0191.212)]. Subspaces of this large superspace (\(X\) typically is not dense in \(h(X,S)\)) were taken up in [the author, Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 15(1972), 111-119 (1973; Zbl 0245.54002)]. The present paper deals again with subspaces of \(h(X,S)\), this time in relation to almost (super- )compactness (\(X\) is almost supercompact relative to \(S\) iff every open cover of \(X\) with members of \(X\) has two members whose union is dense in \(X\); and \(X\) is almost compact if every open cover of \(X\) has a finite subcollection with dense union). In particular, a subspace \(a(X,S)\) is selected which turns out to be ultradense in \(h(X,S)\) and which is important for supercompactness properties of \(X\). Several other subspaces of \(h(X,S)\) are likewise studied. This leads to generalizations of \(H\)- closed extensions as defined by \textit{J. Flachsmeyer} [Math. Z. 94, 349- 381 (1966; Zbl 0147.415)] and generalized by \textit{K. Császár} [Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Math. 19(1976), 49-61 (1977; Zbl 0327.54016)].
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 638239
zbMATH Keywords: hyperextension
zbMATH Keywords: \(H\)-closed centred system
zbMATH Keywords: sieve
zbMATH Keywords: almost supercompact
zbMATH Keywords: almost compact
zbMATH Keywords: \(H\)-closed extensions
reviewed by: Cor Baayen
MaRDI profile type: Publication
cites work: Wallman-type compactifications and superextensions
cites work: Q3866785
cites work: Zur Theorie der H-abgeschlossenen Erweiterungen
cites work: Q5585144
cites work: On $H$-closed extensions of topological spaces
description: scientific article; zbMATH DE number 638239