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)
[edit] 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.
- Baire space is the most common setting for descriptive set theory.
- Used instead of R for these reasons:
- The fact that R is connected is a handicap (e.g. it means images of open sets under continuous functions are too simple).
- 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.
- 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)
