Games and metrisability of manifolds

Abstract

Using topological games we investigate connections between properties of topological spaces and their spaces of continuous functions with the compact-open topology. This leads to new criteria for metrisability of a manifold. We show that a manifold M is metrisable if and only if a winning strategy applies to certain topological games played on C_{k}(M). We also show that M is metrisable if and only if C_{k}(M) is Baire, and even if and only if it is Volterra.