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On the cardinality of certain Hausdorff spaces. (English) Zbl 0766.54002

Summary: We prove (in ZFC) the following theorem. Assume \(\kappa\) is an infinite cardinal, \(X\) is a Hausdorff space such that every subspace \(Y\) of \(X\) is the union of \(\kappa\) compact subsets of \(Y\). Then \(X\) has cardinality at most \(\kappa\).

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Keywords:

ZFC; cardinality
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References:

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