Talk:Polish space

Redirect to Topology glossary

Given that the topology glossary does not in fact include a definition of a polish space, it seemed rather silly to replace this article with a redirect. —The preceding unsigned comment was added by 62.253.128.13 (talk • contribs) 10:43, 26 February 2005 (UTC)

Each Polish Space is a Baire Space

according to the Wikipedia article on Baire Spaces. So "Baire Space" cannot be an example for a Polish Space.

I'm unfamiliar with both notions, so I won't edit the page myself... —The preceding unsigned comment was added by 141.89.48.21 (talk • contribs) 10:03, 9 November 2006 (UTC)

You've run into an unfortunate ambiguity. Some mathematicians use the phrase "a Baire space" to describe any space satisfying the Baire category theorem. Others use the term "Baire space" (no "a") to mean one particular space (namely all functions from the naturals to the naturals, given the product topology). In this article, the second sense is intended. --Trovatore 19:46, 9 November 2006 (UTC)[reply]

Typesetting problem

I've just corrected G_\delta to G_δ. I've tried at first

[[G-delta_set|<math>G_\delta</math>]] subset

which produces $G_{\delta }$ subset. Why this is working wrong? --Kompik 15:49, 18 January 2007 (UTC)[reply]

Well, it seems alright now. I should have made a screenshot -- otherwise nobody believes me that when I've included it for the first time, it has produced something strange. :-) --Kompik 06:30, 22 September 2007 (UTC)[reply]

Converse of Alexandrov's Theorem

I've stumbled upon this result in Arveson. It was mentioned without proof and Bourbaki was included as the reference, therefore I've included both these books into references. —The preceding unsigned comment was added by Kompik (talk • contribs) 18:19, 28 January 2007 (UTC).[reply]

This is also proved in Kechris thm 3.11 Miaoku (talk) 15:56, 19 November 2007 (UTC)[reply]

Confusing statement in the article

This statement doesn't make sense to me:
(Cantor-Bendixson theorem) If X is Polish then any closed subset of X can be written as the disjoint union of a perfect subset and a countable open subset.
By this statement, the set Q of the rationals (i.e. a closed subset of the reals) is the disjoint union of a perfect subset and a countable open set. But there are no countable open subsets in the reals, so this implies that Q is a perfect subset (i.e. a closed set with no isolated points). This is false because every point in Q is an isolated point. Timhoooey 03:02, 14 October 2007 (UTC)[reply]

Good catch; the "open" was nonsense (I guess it's open in the relative topology induced on the subset, but that's not how the statement would ordinarily be read). However your example is wrong -- Q is not closed, and no point in Q is isolated. --Trovatore 03:07, 14 October 2007 (UTC)[reply]

Oops... you're right. 5+ straight hours of studying measure theory will do that to you. I was thinking that countable implies all isolated (as in the naturals) and that Q is closed in itself, not in the reals. Time to take a break. Timhoooey 03:50, 14 October 2007 (UTC)[reply]

Definition of a Lusin space

Out of my element here, but should it not be "X is Lusin if it admits a finer Polish topology" instead of a "weaker" Polish topology? —Preceding unsigned comment added by 83.97.200.245 (talk) 13:08, 23 August 2009 (UTC)[reply]

Image request

User:Hyacinth has added a template to this talk page requesting that an image be added to the article.

I'm not against adding an image if a useful one can be found, but I'm against adding an image just to add an image. Hyacinth, can you say what sort of image you have in mind?

When I think of Polish spaces, I think of Baire space, Cantor space, the reals, and maybe S² (not sure why the last one; I'm just reporting the pictures that pop into my head). Of these, the most "typical" one, in the sense of being used as a default Polish space by descriptive set theorists, is probably Baire space, and it doesn't lend itself to a pretty picture. None of the others seem to especially stand out as champions to carry the banner of Polish spaces. --Trovatore (talk) 01:26, 18 November 2017 (UTC)[reply]

Good thing I added a template that reads, "It is requested that an image or photograph be included in this article to improve its quality." and not a template that reads, "It is arbitrarily requested that cruft be added to this article." Hyacinth (talk) 19:21, 19 November 2017 (UTC)[reply]

I never meant to be confrontational, and I apologize if it came across that way. My point is just that there may not actually be an image that would improve the article's quality.

On the other hand, maybe there is. You did get me thinking about it. The Cantor space is in some sense an archetype of a perfect Polish space (one without isolated points), and if there were a good image showing the middle-thirds version, it might be worth adding.

The tricky part would be the captioning. Too much explanation and it'll just be confusing; too little and it won't be clear what the image has to do with the article. There may not (or may) be a middle ground where it's useful. --Trovatore (talk) 21:20, 19 November 2017 (UTC)[reply]