Variations of selective separability II: discrete sets and the influence of convergence and maximality

Abstract

A space $X$ is called selectively separable(R-separable) if for every sequence of dense subspaces $(D_n : n\in\omega)$ one can pick finite (respectively, one-point) subsets $F_n\subset D_n$ such that $\bigcup_{n\in\omega}F_n$ is dense in $X$. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called \emph{d-separable} if it has a dense $\sigma$-discrete subspace. We call a space $X$ D-separable if for every sequence of dense subspaces $(D_n : n\in\omega)$ one can pick discrete subsets $F_n\subset D_n$ such that $\bigcup_{n\in\omega}F_n$ is dense in $X$. Although $d$-separable spaces are often also $D$-separable (this is the case, for example, with linearly ordered $d$-separable or stratifiable spaces), we offer three examples of countable non-$D$-separable spaces. It is known that d-separability is preserved by arbitrary products, and that for every $X$, the power $X^{d(X)}$ is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite $X$, the power $X^{2^{d(X)}}$ is not D-separable. However, for every $X$ there is a $Y$ such that $X\times Y$ is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming ${\mathfrak d}=\mathfrak c$) there exists a maximal regular countable selectively separable space, and that (in ZFC) every maximal countable space is D-separable (while some of those are not selectively separable). However, no maximal space satisfies the natural game-theoretic strengthening of D-separability.