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

A space X is called selectively separable (R-separable) if for every sequence of dense subspaces ( D n : n ∈ ω ) one can pick finite (respectively, one-point) subsets F n ⊂ D n such that ⋃ n ∈ ω 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 d-separable if it has a dense σ-discrete subspace. We call a space X D-separable if for every sequence of dense subspaces ( D n : n ∈ ω ) one can pick discrete subsets F n ⊂ D n such that ⋃ n ∈ ω 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 × Y is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming d = 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.