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)

Probably this could be a reference, at least for a part of the claims: A.S. Kechris, "Classical descriptive set theory", Springer 1995. Boris Tsirelson (talk) 08:05, 24 November 2008 (UTC)

Almost open[edit]

In some books (Bourbaky, "General Topology" and Kelley, "Geneal Topology"), the sets with the property of Baire are called "almost open": can somebody tell me if it is a common terminology, and if we should add it to the text of the article? Manta (talk) 17:37, 9 January 2009 (UTC)

I have never heard it, but as a general rule I would say that any topological nomenclature attested in both Bourbaki and Kelley is worth at least a mention. I might word it in such a way as to avoid giving the impression that it's a common usage (unless someone pops up to say that it is).
On a side note, not really related to the article, I have to say that I don't like this terminology much and wouldn't like to see it catch on. Sets with the p.o.B don't have to be very much at all like open sets. Any "reasonably definable" set has the p.o.B. --Trovatore (talk) 20:02, 9 January 2009 (UTC)

Sigma algebra[edit]

Shouldn't we mention that the family of sets with the property of Baire is a σ-algebra? --Manta (talk) 13:08, 13 January 2009 (UTC)