Compactness properties defined by open-point games

Abstract

Let S be a topological property of sequences (such as, for example, “to contain a convergent subsequence” or “to have an accumulation point”). We introduce the following open-point game OP(X,S) on a topological space X. In the nth move, Player O chooses a non-empty open set Un⊆X, and Player P responds by selecting a point xn∈Un. Player P wins the game if the sequence {xn:n∈N} satisfies property S in X; otherwise, Player O wins. The (non-)existence of regular or stationary winning strategies in OP(X,S) for both players defines new compactness properties of the underlying space X. We thoroughly investigate these properties and construct examples distinguishing half of them, for an arbitrary property S sandwiched between sequential compactness and countable compactness.