Talk:Property of Baire

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)

message: 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)
thread_title:  Sigma algebra 

message: :The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 
The result of the move request was: No consensus.  No such user (talk) 12:25, 18 January 2023 (UTC)
replies: text: Property of Baire →  – The latter is both more concise and more common in the literature (e.g., 1230 results vs 4280 results on Google Scholar). 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 18:47, 16 December 2022 (UTC)

text: Skeptical.  "Baire property" seems less certain to be talking about this exact notion than does "property of Baire".  For example I think I've seen BP used to describe the proposition that every set of reals has the p.o.B (inconsistent with AC but follows from AD).  So we'd need to do at least a sample of the hits to disambiguate. In any case both formulations should be mentioned in the text. --Trovatore (talk) 19:03, 16 December 2022 (UTC)
replies: text:   We'll need a reliable source for this and evidence that it's widespread. I'm fine with mentioning 'also called "property of Baire"' at the beginning in the lede. 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:08, 16 December 2022 (UTC)
replies: text: Um, hold on, you're the one who wants to change it; you should be making the active case. --Trovatore (talk) 19:10, 16 December 2022 (UTC)
replies: text:  I already did. 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:11, 16 December 2022 (UTC)
replies: text: Well, I remain skeptical.  Let's see what others think. --Trovatore (talk) 19:21, 16 December 2022 (UTC)
replies: text:   Why? 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:26, 16 December 2022 (UTC)
replies: text: I have said why.  Let's see what others think. --Trovatore (talk) 19:27, 16 December 2022 (UTC)

text: Support From Google ngrams, Baire property seems to be much more common than property of Baire in English books published after c. 1960. This is also the only article that is not a work title or a tangible property with a title of the form Property of X instead of X property (Special:PrefixIndex/Property of). –LaundryPizza03 (dc̄) 17:51, 17 December 2022 (UTC)
replies: text: These are a bit unreliable because they only show you how often the words appear together, not how often with a given meaning. --Trovatore (talk) 18:19, 17 December 2022 (UTC)
replies: text: You have shown that there are at least two definitions, but not that there is a meaning unrelated to the article topic. –LaundryPizza03 (dc̄) 00:00, 18 December 2022 (UTC)
replies: text:  I'm not "showing" anything, just pointing out the unreliability of ghit comparisons. --Trovatore (talk) 01:30, 18 December 2022 (UTC)
replies: text:  Can you post examples of "Baire property" being used with a different meaning? 2001:569:7F64:B900:8901:935D:D788:3F85 (talk) 19:30, 26 December 2022 (UTC)

text: Relisting comment: for a clearer consensus – robertsky (talk) 14:04, 27 December 2022 (UTC)

text: Note: WikiProject Mathematics has been notified of this discussion. – robertsky (talk) 14:04, 27 December 2022 (UTC)
replies: text: Looking in more detail at some of the articles in the Google scholar list above, I see that many of them do use "Baire property" to mean the same as "property of Baire" (but not all).  So it's not entirely clear what the best thing would be. PatrickR2 (talk) 03:38, 30 December 2022 (UTC)

text:  On functions having the property of Baire says "Recall that a function has the property of Baire if its preimages of open sets have the property of Baire. A set has the property of Baire if it differs symmetrically from an open set by a meager amount." So "property of Baire" is also overloaded sometimes. 2001:569:7FE4:FD00:5C44:407D:AB66:CF88 (talk) 20:20, 29 December 2022 (UTC)
replies: text: True.  Although we have been discussing various properties of sets here.  In your last example you illustrate that "property of Baire" can be used both to refer to some types of functions, and to some type of sets.  But that is not so confusing if the context is known (sets versus functions). PatrickR2 (talk) 03:44, 30 December 2022 (UTC)

text: Support. I agree with what D. Lazard said. Yes, it doesn't solve the confusion, but property of Baire has the same issues and is less standard, so I think moving to a more standard name will still be a better option. Perhaps a disambiguation page would help the situation afterwards. Pear1020 (talk) 23:12, 9 January 2023 (UTC)
replies: text: Is it less standard?  Not in my experience. --Trovatore (talk) 23:17, 9 January 2023 (UTC)

text: Do you have any data to support that?  It seems that "property of Baire" is equally or more standard in the mathematical literature. PatrickR2 (talk) 05:10, 10 January 2023 (UTC)
thread_title:  Requested move 16 December 2022