On M-separability of countable spaces and function spaces

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of w^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.Comment: 7 page

Remarks on countable tightness

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelof property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are further explored.Comment: Extended from 12 pages to 23 pages. Newly extended to 27 page

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 is an element of omega) one can pick finite (respectively, one-point) subsets F(n) subset of D(n), such that boolean OR(n is an element of 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 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 is an element of omega) one can pick discrete subsets F(n) subset of D(n) such that boolean OR(n is an element of 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(2d(X)) is not D-separable. However, for every X there is a Y such that X 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

On Some Questions About Selective Separability

CH implies that selective separability is not preserved by finite powers (solving [3, Problems 3.7 and 3.9]). In ZFC, selective separability does not imply H-separability (solving [4, Problem 34])