Source on smooth equivalence relations under continuous reducibility? This question was asked and bountied at MSE, but received no answer.
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.)
 A: 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!
A: 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. 


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*Coskey, Amuel; Hamkins, Joel David; Miller, Russell, The hierarchy of equivalence relations on the natural numbers under computable reducibility, DOI:10.3233/COM-2012-004, Computability 1, No. 1, 15-38 (2012). ZBL1325.03049.


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. 
