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

Juhász, István; Szentmiklóssy, Zoltán

Topology Appl. 155, No. 4, 277-281 (2008).

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.

Reviewer: David B. Gauld (Auckland)

MSC:

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

Keywords:

compact space; discrete subspace; \(d\)-separable space; power of a space; \(C_p(X)\)

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