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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)
##### Keywords:
$$\tau$$-extendability