Variations of selective separability
Let \(X\) be a Tychonoff space, \({\mathcal D}\) denote the family of dense subsets of \(X\), and \(C_p(X)\) be the space of continuous real valued functions on \(X\) with the topology of pointwise convergence. We say that \(X\) is \(M\)- \((R\)-)separable in case, for every countable sequence \(\{D_n\}\) with \(D_n\in{\mathcal D}\), there is a finite \(F_n\subset D_n\) (point \(p_n\in C_n)\) for each \(n\) such that \(\cup \{F_n\} (\{p_n\})\) belongs to \({\mathcal D}\). In the present paper these and related properties are studied with the object of extending earlier work by \textit{A. Bella, M. Bonanzinga, M. V. Matveev}, and \textit{V. V. Tkachuk} [Topol. Proc. 32, 15--32 (2008; Zbl 1165.54008)]. The authors show, among many other things, that (i) \(C_p(X)\) is separable and has countable fan tightness if, and only if, \(C_p(X)\) is \(M\)-separable; (ii) if \(C_p(X)\) is \(R\)-separable, then \(C_p(X^n)\) is \(R\)-separable for each finite \(n\) and \((C_p(X))^\omega\) is \(R\)-separable; (iii) if we assume the continuum hypothesis and denote its cardinality by \(c\), then \(2^c\) contains a countable dense \(R\)-separable subspace. A number of open questions are also posed.
scientific article; zbMATH DE number 5546608
scientific article