Talk:Baire space

Definition
I've seen that several authors define the Baire space to be NN, i.e. the irrationals. Might need a disambiguation page eventually. Revolver

Yes, I've seen that too (but only after I started this page!) For now, I'll just add the other definition, but we may well end up splitting them eventually. -- Toby Bartels 03:31, 11 Dec 2003 (UTC)

I'm concerned about the following example:


 * 1) Every space with an isolated point is a Baire space.  (Suppose x is an isolated point of X, and X is the countable union of nowhere dense subsets, then x is a member of a nowhere dense set, which is a contradiction, since the closure of any set containing x has interior at least equal to the singleton {x}, possibly more.)
 * 2) Consider the plane R2, and the subspace X which is the rational numbers Q union the isolated point {(0,1)}.  By the above, since X has an isolated point, it must be a Baire space.  Yet, if we take the singletons { {(q,0)} : q in Q}, then each of these is nowhere dense (they are closed, and has empty interior), so by one of the equivalent formulations of the definition, their union must have empty interior, which it clearly doesn't.  (This example should actually work for any non-Baire space by adding an isolated point to it.)

I think the definitions need to be checked closely, as this seems to be a counterexample. Revolver

I hope this is all straightened out now. I think this definition (every non-empty open set is non-meagre) is correct. In any case, I don't think it's immediately clear to a casual reader that all 4 conditions really are equivalent, and since the proofs are straightforward and short, this might not be a bad thing to have, as well. Revolver

I agree that your counterexample is sufficient to disqualify: as an equivalent definition. Would you agree to delete this definition? Missing the requirement that the nowhere dense set must contain no points isolated in the larger space, this erroneous definition has precipitated the misconception that "any space with an isolated point is a Baire space" as discussed below. A nowhere dense set with no points isolated in the larger space is just a special case of a set with empty interior. Howard McCay (talk) 06:52, 21 June 2013 (UTC)
 * The interior of every union of countably many closed nowhere dense sets is empty.

I think the main modern definition should be: the union of any countable collection of sets with empty interior has empty interior. I think the bottom equivalent definition should read: I do not see the need to require that the sets be closed for these definitions to work. Perhaps you can prove that if these definitions are given for closed sets only, then they can be extended to equivalent assertions applying to sets that are not necessarily closed. So maybe the closed set definitions are in this sense stronger. To me, putting unnecessary qualifications into a definition only makes the definition less clear. Howard McCay (talk) 06:52, 21 June 2013 (UTC)
 * Whenever the union of countably many subsets of X has an interior point, then one of the subsets must have an interior point.

Merged with Baire category theorem
I merged this page with Baire category theorem because I think both articles are too short and too closely related to be discussed on separate pages. I killed the purple boxes around the definitions and inserted some standard headings. Although the boxes looked nice I think the headings yield a better visual structure and standard heading like Definition and Examples provide the reader with faster access to relevant information. I tried to reorder and expand the material a bit to make it clearer but I am not sure I have succeeded. Anyway I intend to spend some more time working on this page the next few days. MathMartin 21:29, 27 Feb 2005 (UTC)

Splitting the set theory part into its own article
As of now, this article has to sections, one about Bair space in topology, and another one, much smaller, about Bair space in set theory. I wonder if it would be a good idea to split the second into Baire space (set theory) and put a link to it at the top of the article, like this:


 * ''For the set theory concept, see Bair space (set theory).

Would that be agreeable? Oleg Alexandrov

American v. British spelling
I understand the principle that the original author's spelling preferences should be kept, but I got to this page via a link from meager set, and then couldn't find "meager" in the page. So I've kept "meagre" as the primary spelling in deference to the principle, but added the Yank spellings as an aid to the wayfarer. --Trovatore 8 July 2005 22:45 (UTC)
 * See Manual of Style.  There's really no point in using both spellings - it's easy to see that "meager" is the American way of spelling "meagre" (or equally that "meagre" is the British way of spelling "meager"), and including both just clutters up the article.  The general principle is that if there is no reason to use one or the other (true in this case), the first editor's preference should be used.  http://en.wikipedia.org/w/index.php?title=Baire_space&direction=next&oldid=834287 uses "meagre", so we go with British spelling.  Hairy Dude 22:59, 15 January 2007 (UTC)

Question
An alternative definition of baire space: "Every intersection of countably many dense open sets is dense."

This seems unreasonable to me. Take for example the collection of sets $$ \ A_p=\mathbb Q + \frac{\pi}{p}$$, (where p is rational) the shift of the rationals by a non-rational number. This collection is pairwise disjoint, and therefore $$ \bigcap_p A_p=\emptyset $$ which is not dense. Am I wrong? --Yohai.bs 10:53, 20 July 2007 (UTC)


 * but those sets are "too small": they ain't open.  Mct mht 06:02, 24 July 2007 (UTC)


 * that is why openness is crucial (the intersection of the rationals and irrationals in $$R$$ is not dense but this does not imply that $$R$$ is not a Baire space).

Topology Expert (talk) 04:44, 31 October 2008 (UTC)

Any space with an isolated point is a Baire space
I just removed the sentence It should be pointed out that any space containing an isolated point is a Baire space, because an isolated point is its own interior. because it is false. It is true that an isolated point is its own interior, but that does not make the space a Baire space! If you take a non-Baire space (like the positive rational numbers) and add a single isolated point (like -1), then you will NOT get a Baire space! (The original space is an open subset of the new space with the additional point.  If the new space was a Baire space, the original one would also be one...) --131.234.106.197 (talk) 11:27, 19 May 2008 (UTC)

Very large, is it?
"Baire space is a topological space which, intuitively speaking, is very large" — really? A one-dimensional closed interval, is it "very large"? But wait; a finite set of points (even a single point), are they "very large"? What a strange intuition... Boris Tsirelson (talk) 11:02, 17 September 2013 (UTC)


 * I agree, it is not a great description. I rewrote the lead to be a bit more concrete. See if you find it more reasonable. --Mark viking (talk) 12:58, 17 September 2013 (UTC)


 * Better; but I still wonder: finite sets of points (even a single point), are they "very large"? Boris Tsirelson (talk) 20:40, 17 September 2013 (UTC)