On the open-open game

We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II does or does not have a winning strategy. We investigate the games on trees and Boolean algebras in detail, completely characterizing the game for ω1-trees. An undetermined game is also defined. (In contrast, it is still open whether there is an undetermined game using the definition due to Berner and Juhász.) Finally, we show that various variations on the game yield equivalent games. 0. Introduction. The following game is due to Berner and Juhász [BJ]: two players take turns playing; a round consists of player I choosing a nonempty open set U ⊆ X and player II choosing a point p ∈ U ; a round is played for each ordinal less than ω (more generally, for each ordinal less than some given ordinal α); I wins the game if the set of points II plays is dense; otherwise, II wins. Denote the game by Gp(X). Consider the following modification, which we call the open-open game of length ω: a round consists of I choosing a nonempty open set U ⊆ X and II choosing a nonempty open V ⊆ U ; I wins if the union of II’s open sets is dense in X, otherwise II wins. Denote this game by G(X). (In comparison with Gp(X), G o 0(X) might be a better notation.) Gp(X) is only interesting if X is separable, as otherwise II has a trivial winning strategy. G(X) is interesting for a wider class of spaces, the ccc spaces (if X is not ccc, it is easy to see that II has a winning strategy). Furthermore, the open-open game can be translated into an interesting game 1991 Mathematics Subject Classification: 54A35, 03E35, 54D99. The second author was partially supported by NSF Grant DMS-9100664.