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 ${\displaystyle {\omega }_1}$-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.