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´asz,
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 T3 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, - and !-covering properties on X are shown to be equivalent
to corresponding sequential properties on Cp(X). A strengthening of the
Ramsey property is also introduced, which is still equivalent to a_2 and a_4 in
the context of Cp(X).