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Pseudocompact Whyburn spaces need not be Fréchet. (English) Zbl 1028.54004
Spaces which are determined by almost closed subspaces are called Whyburn spaces; these spaces were introduced by G. T. Whyburn [ibid. 24, 181–185 (1970; Zbl 0197.48602)] and later studied by several authors under the name AP-spaces. The authors give some necessary and sufficient conditions for a \(P\)-space to be Whyburn, show that a linearly ordered topological space is Whyburn iff it is club incomplete and that a linearly ordered Lindelöf \(\Sigma\)-space is Whyburn iff it is first countable. Also, they give examples of a space of uncountable tightness which is a Tychonoff pseudocompact scattered Whyburn space, a weakly Whyburn regular Lindelöf \(P\)-space and a Lindelöf linearly ordered Whyburn space, respectively.
Remark: Conditions for a Whyburn space to be a Fréchet space were given by Ch. E. Aull [Czech. Math. J. 29(104), 178–186 (1979; Zbl 0424.54013)].

MSC:
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D99 Fairly general properties of topological spaces
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54G12 Scattered spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54G10 \(P\)-spaces
54G20 Counterexamples in general topology
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