A pseudocompact meta-Lindelöf space which is not compact
type: Item
instance of: scholarly article
title: A pseudocompact meta-Lindelöf space which is not compact
zbMATH Open document ID: 0589.54030
DOI: 10.1016/0166-8641(85)90091-4
author: Stephen Watson
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: A Tychonoff space is pseudocompact if every countable open filter base has a cluster point. A space is called meta-Lindelöf provided every open cover of the space has a point-countable open refinement. The example constructed in this paper (described in the title) can be compared with a theorem of D. K. Burke and S. W. Davis which states that a Tychonoff, pseudocompact, para-Lindelöf space is compact, where a space is para-Lindelöf provided every open cover has a locally countable open refinement (the proof of this can be found on page 416 in the article by \textit{D. K. Burke} in Handbook of set-theoretic Topology, 347-422 (1984; Zbl 0569.54022)). The example X is nontrivial; it is a subspace of the space \(Y=F\cup G\) which consists of the full \(\omega\)-ary tree F of all functions with domain a countable ordinal and range a subset of the set of natural numbers (with a natural tree topology) and a set g which contains certain countable sequences in F (the points of G are used to get pseudocompactness in analogy with the topology of the well-known space \(\Psi)\). The desired space X is then defined as a subset of Y by transfinite induction.
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 3947173
zbMATH Keywords: pseudocompact meta-Lindelöf space
zbMATH Keywords: pseudo-Lindelöf space
reviewed by: Jerry E. Vaughan
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90091-4
OpenAlex ID: W2047653201
cites work: Q5338726
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description: scientific article; zbMATH DE number 3947173