Games, covering properties and Eberlein compacts

AbstractConsider the following game played in a locally compact space X: at the nth move, K chooses a compact set Kn⊂X, and then P chooses a point pn∉∪{Ki: i⩽n}. We say K wins ifP's points have no limit point in X. We show that X is metacompact (σ-metacompact) if and only if K has a strategy in this game which depends only on P's last move (and the number of the move). As a corollary we obtain a game characterization of Eberlein compact spaces. We also show that if P is allowed to choose compact sets instead of points, then K has a winning strategy if and only if X is paracompact