Characterizations of \(\aleph\)-spaces
type: Item
instance of: scholarly article
title: Characterizations of \(\aleph\)-spaces
zbMATH Open document ID: 0542.54030
author: L. Foged
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Two simultaneous generalizations of metrizable spaces and \(\aleph_ 0\)- spaces, namely P. O'Meara's \(\aleph\)-spaces and J. Guthrie's cs-\(\sigma\)- spaces, are shown to be the same. In fact, it is proved that the following are equivalent for a regular space X: (a) X has a \(\sigma\)- locally finite cs-network. (b) X has a \(\sigma\)-discrete k-network. (c) X has a \(\sigma\)-locally finite cs-network. (d) X has a \(\sigma\)-locally finite k-network. [A collection \({\mathcal P}\) of subsets of X is a k-network (resp. cs-network) if, whenever \(S\subset U\) with S compact (resp. S a convergent sequence) and U open in X, then there is \(P\in {\mathcal P}\) such that \(S\subset P\) (resp. S is eventually in P) and \(P\subset U]\).
Mathematics Subject Classification ID: 54E99
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 3862006
zbMATH Keywords: \(\aleph_ 0\)-spaces
zbMATH Keywords: \(\sigma\)-locally finite cs-network
zbMATH Keywords: \(\sigma\)- discrete k-network
reviewed by: E. Michael
MaRDI profile type: Publication
full work available at URL: pjm.1984.110.59
OpenAlex ID: W1537115342
DOI: 10.2140/PJM.1984.110.59
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description: scientific article; zbMATH DE number 3862006