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On AP and WAP spaces. (English) Zbl 1010.54040

A topological space \(X\) has the property AP (or WAP) if for every nonclosed \(A\subset X\) and every (or some, resp.) \(x\in \overline {A}\smallsetminus A\) there exists some \(B\subset A\) with \(\overline {B}\smallsetminus A =\{x\}\). Among others it is shown that (1) there are countable AP spaces \(X\) and \(Y\) (moreover, \(Y\) is Fréchet-Uryson) such that \(X\times (\omega +1)\) is not WAP and \(Y\times (\omega +1)^2\) is not AP; (2) the space \(C_p(\kappa)\) is WAP but not AP whenever \(\kappa \) is a regular \(\omega \)-inaccessible cardinal.

MSC:

54F99 Special properties of topological spaces
54C35 Function spaces in general topology
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