On the instability of a topological game related to consonance

Abstract

We consider separable metric spaces with the property that Player One has no winning strategy in the topological game where Player One plays an open k-cover of B and Player Two picks an element of the k-cover in each round. Here Player One wins when the choices of Player Two fail to cover B. We show that the topological sum of two such spaces may no longer have the property. From this we answer some open questions about such spaces as well as about consonant spaces.