×

Selective separability: general facts and behavior in countable spaces. (English) Zbl 1165.54008

A space \(X\) is called selectively separable if, for any sequence \(\{D_n: n\in \omega \}\) of dense subsets of \(X\), we can choose a finite set \(F_n\subseteq D_n\) for each \(n\in \omega\) in such a way that \(\bigcup_{n\in \omega} F_n\) is dense in \(X\); this notion was introduced by M. Scheepers [Quaest. Math. 22, No. 1, 109–130 (1999; Zbl 0972.91026)]. It is known that every space of countable \(\pi\)-weight is selectively separable and, if \(X\) is selectively separable, then all dense subsets of \(X\) are separable. In this paper, the general properties of selective separability together with the behaviour of this notion in some special classes, such as function or countable spaces, are studied.
The following main results are proved: (1) a space \(C_p(X)\) is selectively separable if and only if it is separable and has countable fan tightness (this is a variation of a previous result by Scheepers where the selective separability of \(C_p(X)\) is characterized in terms of \(X\) for second countable spaces [op. cit.]); (2) The space \(\{0,1\}^{\mathfrak d}\) contains a countable dense subspace which is not selectively separable; (3) If \(\mathfrak d=\omega_1\), then there exists a regular maximal countable space \(X\) which is not selectively separable. The paper also contains other interesting results and open questions.

MSC:

54D65 Separability of topological spaces
54C35 Function spaces in general topology
54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Citations:

Zbl 0972.91026
PDFBibTeX XMLCite