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On star-compact spaces with a \(G_\delta\)-diagonal. (English) Zbl 0911.54020
Summary: The authors introduce a new concept – \(\tau\)-extendability of topological properties, and prove that certain compactness-type properties are \(\tau\)-extendable for uncountable \(\tau\). Answering D. B. Shakhmatov’s question we present for any cardinal \(\lambda\) an example of a pseudocompact space \(X\) with \(G_\delta\)-diagonal and \(| X|\geq \lambda\). Under CH this space \(X\) can be made 2-pseudocompact which is stronger than being pseudocompact. A ZFC example of a 2-pseudocompact space which has no dense relatively countably compact subspace is also given.

MSC:
54D30 Compactness
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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