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On \(d\)-separability of powers and \(C_p(X)\). (English) Zbl 1134.54002

A space is \(d\)-separable if it has a dense set representable as a countable union of discrete subsets. Let \(\hat s(X)\) be the smallest cardinal \(\lambda\) such that \(X\) has no discrete subspace of size \(\lambda\). If \(X\) is T\(_1\) and \(\kappa\) is a cardinal with \(\hat s(X^\kappa)>d(X)\) then \(X^\kappa\) is \(d\)-separable, while if \(X\) is compact T\(_2\) then \(\hat s(X^2)>d(X)\). Consequently if \(X\) is T\(_1\) then \(X^{d(X)}\) is \(d\)-separable while if \(X\) is compact T\(_2\) then \(X^\omega\) is \(d\)-separable. Examples are also given of spaces which are not \(d\)-separable.

MSC:

54B10 Product spaces in general topology
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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References:

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