On Ramsey properties, function spaces, and topological games

Abstract

An open question of Gruenhage asks if all strategically selectively separable spaces are Markov selectively separable, a game-theoretic statement known to hold for countable spaces. As a corollary of a result by Berner and Juh$\acute{a}$sz, we note that the strong version of this statement, where the second player is restricted to selecting single points in the rather than finite subsets, holds for all $T_3$ spaces without isolated points. Continuing this investigation, we also consider games related to selective sequential separability, and demonstrate results analogous to those for selective separability. In particular, strong selective sequential separability in the presence of the Ramsey property may be reduced to a weaker condition on a countable sequentially dense subset. Additionally, $\gamma$- and $\omega$- covering properties on $X$ are shown to be equivalent to corresponding sequential properties on $C_p(X)$. A strengthening of the Ramsey property is also introduced, which is still equivalent to $\alpha_2$ and $\alpha_4$ in the context of $C_p(X)$.