Talk:Baire space (set theory)

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Before my edit, this page suggested that the topology on Baire space was an afterthought and not very important. That's not true; its topology (or at least the Borel structure derived therefrom) is fundamental to applications in descriptive set theory. --Trovatore 1 July 2005 00:53 (UTC)

Work needed on this page

User:Light current has a good point that this article needs better organization, but I didn't think adding the header "description" right after the first sentence helped much. I'm not sure what will, though. Let me lay out a few points currently missing and see if someone sees how to put it together into a more coherent picture.

1. Baire space is the most common setting for descriptive set theory.
2. Used instead of R for these reasons:
   1. The fact that R is connected is a handicap (e.g. it means images of open sets under continuous functions are too simple).
   2. A related point: Baire space is zero-dimensional, so powers of Baire space are also zero-dimensional (and in fact homeomorphic to Baire space) -- this means that when we code pairs (or tuples) of elements of Baire space by a single element of Baire space, we don't have awkward topological considerations to deal with.
   3. We could use Cantor space instead, and sometimes do, but it's compact, which is sometimes limiting (for example, it has only countably many clopen subsets). Baire space, in contrast, is not locally compact.

The intro should summarize these points, which can then be dealt with in detail; that could lead to natural headers. --Trovatore 03:12, 10 November 2005 (UTC)

Baire space-based construction of nonstandard model of arithmetic

If one quotients N^N by the relation declaring two sequences to be equivalent if they agree on a set of indices which is a member of a fixed ultrafilter, then one obtains a non-standard model of arithmetic. It is my understanding that this is equivalent to Skolem's original construction of such models, but I have not seen it in the original documents. It may be worth mentioning it here as one of the applications. Tkuvho (talk) 19:27, 9 August 2010 (UTC)

It's the wrong topology, though, so this may not be too relevant. Tkuvho (talk) 19:38, 9 August 2010 (UTC)

Huh. That's interesting. "wrong topology" -- is the other topology the box topology? Oh, but of course it is, it would have to be, for the indicator set to belong to an ultrafilter. Its worth mentioning somehow, anyway. 67.198.37.16 (talk) 18:52, 19 September 2015 (UTC)

Who calls it B?

I do not believe I have seen the notation B for the Baire space anywhere but Wikipedia. Certainly neither of the refs calls it that. Moschovakis calls it ${\mathcal {N}}$, which is probably slightly idiosyncratic, and I believe Kechris calls it either ω^ω or ^ωω, which are both more standard.

My suggestion would be to use ω^ω, which I think is probably the most common usage among descriptive set theorists. --Trovatore (talk) 01:59, 5 September 2015 (UTC)

A nicer homeomorphism

There is a far nicer homeomorphism between the irrationals and the Baire space than the cited one using continued fractions. I think that the continued fractions is rather difficult to visualize or understand how it works --it is rather technical. A far more intuitive homeomorphism is constructed as follows:

Partition the irrationals into countably many pairwise disjoint open intervals with rational endpoints:

.... -3  -2   -1       0       1     2  3 ....
....(  )(  )(    )(        )(     )(  )( )....

This is the first branching from the root of the Baire tree to the first level.

Then for each open interval, repeat the splitting into intervals with rational endpoints.

....       -2                 -1                     0                       1         ....
....(               )(                 )(                          )(                 )....
     ...-1  0  1 ...  ...-1   0   1 ...  ...-2  -1   0    1   2 ...  ...-1   0   1 ...
....(...( )( )( )...)(...( )(   )( )...)(...( )(  )(   )(  )(  )...)(...( )(   )( )...)....

This gives you the second level of the Baire tree, where every node from the first level then splits into countably many children.

Continue repeating this construction through $\omega$-many levels. If you are careful with your bookkeeping (Enumerate the rationals q_0,q_1,q_2,... and use the rule that at stage i+1, if q_i has not been used, then one of the cuts has to be at q_i), you can make a cut at every rational number exactly once. This defines the homeomorphism. Each irrational is uniquely determined by a sequence of integers and every sequence of integers uniquely determines an irrational number.

Perhaps someone with more wiki-perience could add this to the page. I think this is far more helpful than the continued fractions nonsense.

37.191.140.227 (talk) 15:27, 10 September 2015 (UTC)IKnowNothingAboutTopology37.191.140.227 (talk) 15:27, 10 September 2015 (UTC)

Yes, this seems like a plausible, but slightly flawed construction. Anyway, its more-or-less "original research", and original research is banned on WP. But this construction is perhaps "obvious" enough that perhaps its been published somewhere already. Its almost but not quite how Cantor talks about things in 1890. (Cantor does what you do, but just splitting into two, rather than countable. Go read his original paper; its interesting; there's a compilation "classical papers on topoogy" or something like that, that you can easily get) Anyway, the point of continued fractions is that they are highly technical, and have a fascinating interaction with the modular group; your construction can be converted into a continued fraction by applying a sequence of hyperbolic rotations on the upper half-plane. (Here's how: at the first level, you rotate to bring your rational endpoints into alignment with 1/n; and then repeat at each level.)67.198.37.16 (talk) 18:30, 19 September 2015 (UTC)

This is not original research. I saw this almost 40 years ago. As far as I know, it's the 'standard' proof. It is exactly the analogue to the proof that the Cantor set is homeomorphic to $2^\omega$. I am sure continued fractions are very interesting, but the proof I presented is much easier to understand with only a basic understanding of topology. Take it or leave it. And, no, the proof is not flawed. 37.191.140.227 (talk) 09:16, 2 October 2015 (UTC)IKnowNothingAboutTopology37.191.140.227 (talk) 09:16, 2 October 2015 (UTC)

A reference would be nice. The ASCII-art diagram is rather hard to view and understand.67.198.37.16 (talk) 19:22, 28 January 2016 (UTC)

How is N^N a Polish space?

The section Properties list this as property 1.:

"It is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points. As such, it has the same cardinality as the real line and is a Baire space in the topological sense of the term."

It would be a very useful addition to the article if it were mentioned how this is possible.

In other words: What is an example of a complete metric space structure on N^N that defines the correct topology?

I hope someone knowledgeable about this subject can add this important fact. — Preceding unsigned comment added by 2601:200:c082:2ea0:6115:5f27:a5ac:8057 (talk • contribs) 15:27, 9 April 2024 (UTC)

It's not hard; pretty much anything reasonable that you try will work. For example you could map the integers into [0, 1], then sum up the difference in the nth coordinate of two sequences (after mapping into [0, 1]) divided by 2ⁿ. But I'd need to find it in a reliable source in order to add it. Probably out there somewhere. --Trovatore (talk) 16:09, 9 April 2024 (UTC)

Unclear statement

In the section Relation to the real line this passage appears:

"From the point of view of descriptive set theory, the fact that the real line is connected causes technical difficulties. For this reason, it is more common to study Baire space."

More common than what ????? This is never stated.

And what "technical difficulties" are referred to here?

Also: How is it possible that the Alexander-Urysohn theorem is not mentioned in this article?

The theorem states "The Baire space ℕ^ℕ is the unique, up to homeomorphism, non-empty Polish zero-dimensional space for which all compact subsets have empty interior." (Theorem 3.3 in https://webusers.imj-prg.fr/~dominique.lecomte/Chapitres/3-The%20Cantor%20and%20the%20Baire%20spaces.pdf.)

I hope that someone who is knowledgeable about this subject can add this important fact to the article. — Preceding unsigned comment added by 2601:200:C082:2EA0:1C91:8A6:21B7:9051 (talk) 17:23, 11 April 2024 (UTC)