Proper forcing axiom and selective separability

摘要

We 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 C_P(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 Frechet 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 Frechet spaces have π-weight at most ω_1.