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A generalization of F-spaces and some topological characterizations of GCH. (English) Zbl 0535.54023

E. van Douwen, using earlier results of the reviewer, established several topological characterizations of the continuum hypothesis involving topological properties of F-spaces X for which \(| C^*(X)| =c\). Let \(\alpha\) be an infinite cardinal. The author defines an \(F_{\alpha}\)-space to be a Tychonoff space X in which the union of \(<\alpha\) cozero-sets of X is \(C^*\)-embedded in X (thus F-spaces are \(F_{\aleph_ 0}\)-spaces). She then characterizes the cardinal equality \(2^{\alpha}=\alpha^+\) by using topological properties of \(F_{\alpha}\)-spaces, thereby generalizing the work of the reviewer and van Douwen.
Reviewer: R.G.Woods

MSC:

54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
54C45 \(C\)- and \(C^*\)-embedding
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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