A pseudocompact meta-Lindelöf space which is not compact
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.
scientific article; zbMATH DE number 3947173
scientific article