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In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since everything boils down to the number of classes. However, the situation seems more interesting when we restrict attention to continuous reducibility. I'd like to know more about the structure of smooth equivalence relations with respect to continuous reducibility; what is a good source on this topic? (I'm happy to restrict to equivalence relations on Baire space if that would help, but in general I'm interested in arbitrary Polish spaces.)

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    $\begingroup$ Let me ask a more basic question. Every $G_{\delta}$ equivalence relation on a Polish space is smooth. What is an example of a smooth Borel equivalence relation which is not potentially-$G_{\delta}$? (Potentially-$\Gamma$ means that it is in the pointclass $\Gamma$ for some finer Polish topology inducing the same Borel structure.) $\endgroup$
    – Burak
    Commented Nov 6, 2019 at 23:18
  • $\begingroup$ @Burak I like that question very much! $\endgroup$
    – Noah Schweber
    Commented Nov 7, 2019 at 1:12
  • $\begingroup$ The following paper might be of some interest. On the complexity of Borel equivalence relations with some countability property. $\endgroup$
    – Burak
    Commented Dec 17, 2019 at 10:46

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Seems like this structure must be pretty complicated. For example, consider Brownian motion $\{W_t\}_{t\ge 0}$ with the equivalence relations $$t\sim_\omega s\iff W_t(\omega)=W_s(\omega).$$ Here $\omega\in\Omega$ is an outcome from the sample space, giving a particular path of Brownian motion.

This $\sim_\omega$ is smooth. It seems that these equivalence relations should almost surely not be continuously reducible to eachother, i.e., for almost all $(\omega_1,\omega_2)\in\Omega\times\Omega$ there should not be any continuous $f$ with $$t\sim_{\omega_1} s\iff f(t)\sim_{\omega_2}f(s).$$ However I don't have a proof of that... just a thought!

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  • $\begingroup$ Sorry, I'm not familiar with Brownian motion - can you say a bit about what $W_t(\omega)$ (etc.) is (or point me to a text in case that's too much material to quickly summarize)? $\endgroup$
    – Noah Schweber
    Commented Nov 6, 2019 at 23:09
  • $\begingroup$ @NoahSchweber It's a non-differentiable-but-Hölder-continuous-function-valued random variable... $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Nov 7, 2019 at 3:02
  • $\begingroup$ @NoahSchweber anyway here's some more about this idea mathoverflow.net/q/345528/4600 $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Nov 9, 2019 at 7:13
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This is more of a comment than an answer, since it is not a perfect fit. But I just thought I would mention the following paper, which is concerned not with continuous reducibility, but computable reducibility.

This is not the same thing, of course, but since computable functions on a countable space can be seen as continuous on the reals, it might be a source of some examples.

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  • $\begingroup$ If we restrict to computable, my "answer" becomes a lot easier to prove too (and the event becomes clearly measurable) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Nov 7, 2019 at 3:12

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