Talk:Property of Baire

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The article says, "If a subset of a Polish space has the property of Baire, then its corresponding Banach-Mazur game is determined." This would indicate that there is one Banach-Mazur game corresponding to that set (presumably where the goal for player II is to land in the set in question), but this makes no reference to the set Y from which moves by the two players are chosen. Is there a specific set Y which is assumed when none is explicitly mentioned? I think some more explanation is needed. Althai 16:24, 4 March 2007 (UTC)

Well, if the set has the property of Baire, then it doesn't really matter how you formulate the Banach-Mazur game; it's going to be determined. The usual answer would be that your Y is the collection of all basic open neighborhoods in the space ("basic open" in some chosen countable basis for the topology). The choice of a countable basis is also arbitrary, but unimportant, and in fact you don't need to do it at all, really; the players could play arbitrary open sets if they wanted and it wouldn't change the game in any important way. The only thing that would change is that there would no longer be a direct coding of the game as a game played on the natural numbers. --Trovatore 19:06, 4 March 2007 (UTC)

OK, so I took a look at the Banach-Mazur game page and now I see what you're talking about. That's not quite the way I think of the game. The way I think of it, the players can play arbitrary open sets, not just ones in a collection Y with the property about closures. But, they're required to ensure at each move that the closure of the set they play is a subset of their opponent's last move. In any case, as I say, it doesn't matter for the purposes of this article -- no matter how you fiddle with the Y, the claim made here is still true. --Trovatore 19:13, 4 March 2007 (UTC)

Should the definition here perhaps be rewritten, or at least some mention of this version made? I was introduced to Banach-Mazur games in computability theory, so we generally only use games in Cantor space where moves are the basic clopen sets. Althai 05:56, 5 March 2007 (UTC)