Talk:Property of Baire

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

Changed "property of Baire" to "Baire property"[edit]

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) 18:39, 16 December 2022 (UTC)[reply]

I undid this change. "Property of Baire" is definitely in use, and in any case both terms are mentioned. If you want to move the article, see the procedure at WP:RM and we can discuss it — it's not completely implausible but should be discussed. --Trovatore (talk) 18:41, 16 December 2022 (UTC)[reply]

@Trovatore: "Property of Baire" is definitely in use First, I already noted that. Second, WP:COMMONNAME says: [Wikipedia] generally prefers the name that is most commonly used... the term or name most typically used in reliable sources is generally preferred.

Besides the Google Scholar results, other examples favoring "Baire property":

2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:00, 16 December 2022 (UTC)[reply]

Requested move 16 December 2022[edit]

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)[reply]

Property of Baire → Baire property – 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) This is a contested technical request (permalink). 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 18:49, 16 December 2022 (UTC) — Relisting. – robertsky (talk) 14:04, 27 December 2022 (UTC)[reply]

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)[reply]

@Trovatore: I think I've seen BP used to describe the proposition that every set of reals has the p.o.B 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)[reply]

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)[reply]

@Trovatore: I already did. 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:11, 16 December 2022 (UTC)[reply]

Well, I remain skeptical. Let's see what others think. --Trovatore (talk) 19:21, 16 December 2022 (UTC)[reply]

@Trovatore: I remain skeptical Why? 2001:569:7D87:F00:A987:98B3:6A08:3590 (talk) 19:26, 16 December 2022 (UTC)[reply]

I have said why. Let's see what others think. --Trovatore (talk) 19:27, 16 December 2022 (UTC)[reply]

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 (d c̄) 17:51, 17 December 2022 (UTC)[reply]

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)[reply]

You have shown that there are at least two definitions, but not that there is a meaning unrelated to the article topic. –LaundryPizza03 (d c̄) 00:00, 18 December 2022 (UTC)[reply]

I'm not "showing" anything, just pointing out the unreliability of ghit comparisons. --Trovatore (talk) 01:30, 18 December 2022 (UTC)[reply]

@Trovatore: 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)[reply]

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

Note: WikiProject Mathematics has been notified of this discussion. – robertsky (talk) 14:04, 27 December 2022 (UTC)[reply]

Support. "Property of Baire" is ambiguous and thus confusing, as meaning also "property that the mathematician had". The two meanings are distinguished by the different meanings of "Baire property" and "Baire's property". D.Lazard (talk) 14:43, 27 December 2022 (UTC)[reply]
Unfortunately there are multiple notions of sets having the name of Baire in them: property of Baire, Baire set, Baire space, more? It certainly can be confusing and would take a while for a beginner to grasp the differences. Nevertheless, the phrase "property of Baire" has been used with a very specific meaning for decades and I don't think replacing that with something else will help lessen the confusion regarding the various terms above. See some evidence in my Oppose comment below. PatrickR2 (talk) 21:54, 28 December 2022 (UTC)[reply]
Comment How about Baire set? A Baire set is a certain kind of subset of a topological space: a subset that has the Baire property. Michael Hardy (talk) 18:57, 27 December 2022 (UTC)[reply]
@Michael Hardy: Are you sure about this? Doesn't "Baire set" and "set with the property of Baire" refer to different concepts? They are at least defined differently, and neither of the two articles mentions that the notions are the same. PatrickR2 (talk) 21:16, 28 December 2022 (UTC)[reply]
Oppose I would have to agree with Trovatore here. Independently of any wikipedia conventions about "X property" versus "property of X", the phrase "property of Baire" has been used with a very specific meaning for decades. In particular, the book by Oxtoby, Measure and Category, has a whole chapter dedicated to that notion: https://books.google.com/books?id=Va_aBwAAQBAJ&pg=PA19&dq=oxtoby+%22measure+and+category%22+%22property+of+Baire%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjB8uqcoZ38AhX6NlkFHek9DJYQ6AF6BAgHEAI#v=onepage&q=oxtoby%20%22measure%20and%20category%22%20%22property%20of%20Baire%22&f=false And the phrase "Baire property" does not alway refer to "property of Baire". Looking at a random example from the list in Google scholar provided at the top of this discussion, I see that some authors talk for example of a space having the Baire property just to mean a Baire space, which could be a reasonable use of the phrase "Baire property" in such a context. See for example https://link.springer.com/article/10.1007/s13398-022-01371-w Note that this is a completely different notion than "property of Baire". So it seems to me that keeping "property of Baire" as the main term would be more precise and less prone to confusion.

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)[reply]

@PatrickR2: 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)[reply]

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)[reply]

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)[reply]

Is it less standard? Not in my experience. --Trovatore (talk) 23:17, 9 January 2023 (UTC)[reply]

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)[reply]

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.