Geodesic
On the open-open game
Fundamenta Mathematicae, Tome 145 (1994) no. 3, pp. 205-220 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $ω_1$-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.
DOI : 10.4064/fm_1994_145_3_1_205_220

Peg Daniels 1 ; Kenneth Kunen 1 ; Haoxuan Zhou 1

1 
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Peg Daniels; Kenneth Kunen; Haoxuan Zhou. On the open-open game. Fundamenta Mathematicae, Tome 145 (1994) no. 3, pp. 205-220. doi: 10.4064/fm_1994_145_3_1_205_220

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