Proper forcing axiom and selective separability

AbstractWe continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS+) (see Barman, Dow (2011) [1]). The motivation for studying SS+ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS+ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS+ spaces such that the union fails to be SS+, which contrasts the known result about SS spaces. We also prove that the product of two countable SS+ spaces is again countable SS+. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω1