# Talk:Borel equivalence relation

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Field:  Foundations, logic, and set theory

• Hierarchy of Borel cardinalities
• Finite cardinals, ${\displaystyle \mathbb {N} }$, ${\displaystyle \mathbb {R} }$, ${\displaystyle E_{0}}$
• Dichotomies: No cardinals between ${\displaystyle \mathbb {N} }$ and ${\displaystyle \mathbb {R} }$ (Silver-Harrington). No cardinals between ${\displaystyle \mathbb {R} }$ and ${\displaystyle E_{0}}$ (Harrington-Kechris-Louveau, generalization of Glimm-Effros).
• After that, no longer a linear order
• But there's another local dichotomy between ${\displaystyle E_{0}}$ and ${\displaystyle \ell ^{1}}$ (Hjorth) (Well, almost.)
• Countable equivalence relations
• Equivalence relations induced by Polish group actions
• Equivalence relations reducible to isomorphism on countable structures
• Turbulent actions
• Louveau-Velickovic relations
• Hyperfinite equivalence relations and the union problem.

## Split!

Why the hell does standard Borel space redirect here? Standard Borel spaces (and Kuratowki's theorem) have small connection with Borel equivalence relation and may defintely not be considered as a subtopic of it. 2A01:E35:8B38:8680:D897:F1E7:2D81:E0D (talk) 07:55, 4 July 2017 (UTC)

Probably because this article does define it, and someone wanted the search term to point somewhere. I agree it's not the optimal target in the best of all possible worlds.
You are free to help correct the situation. If there's a better target, you can just point it there (maybe standard probability space? But that's not ideal either).
It's not clear to me that the standard Borel space merits a standalone article; there may not be that much to say about it in and of itself (and Wikipedia is not a dictionary). So maybe the redirect should instead point to an entry in glossary of topology, if you want to write one.
If you think the standard Borel space should have a standalone article, and you can find references to support it, you can write it. Non-logged-in editors can do almost everything registered users can do, but they can't create articles, so you have two choices:
1. Create an account. This is by far the better choice. Registering an account gives an easy way for other editors to contact you about your work. I strongly recommend that you do this if you want to write articles (or even if you don't). Follow this link; it's very easy.
2. But if you really insist on editing from an IP address, you can start an article at Wikipedia:Articles for Creation, and when you're done, ask that someone move it into mainspace for you.
Good luck! Looking forward to seeing your work. --Trovatore (talk) 08:37, 4 July 2017 (UTC)

Oh, it just occurred to me that, since the page standard Borel space already exists, you don't have to be logged in to turn it into an article. I still recommend it very very strongly. It makes conversations much easier when people know they're talking to the same person. --Trovatore (talk) 08:43, 4 July 2017 (UTC)
I created the page (and logged in), but it appears that it might be a good idea to merge with Borel_set#Standard_Borel_spaces_and_Kuratowski_theorems and (maybe) Borel isomorphism. pom (talk) 21:32, 9 July 2017 (UTC)
Looks good. The merge you suggested might be reasonable, though I'm not sure Borel set is the best place for the content.
One other minor annoyance that I'm not sure what to do about. Our content generally talks about standard Borel spaces, or "a" standard Borel space. But up to isomorphism, there is only one nontrivial one. The countable ones are degenerate cases. In my experience, people talk about "the" standard Borel space; that is, the unique-up-to-isomorphism uncountable one.
I think it's a little misleading to talk about different standard Borel spaces. But I don't know what to do about it because I don't know where to find a written source. --Trovatore (talk) 05:28, 10 July 2017 (UTC)