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Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, the behavior of the usual topology and the “condensed” one, align. However, for some quotient spaces, like $\mathbb{R} / \mathbb{Q}$, the usual quotient topology is indiscrete, so every map $f : \mathbb{R} / \mathbb{Q} \to \mathbb{R} / \mathbb{Q}$ is continuous. However, as a condensed set, more structure is preserved. Indeed, the “continuous” maps are exactly those coming from continuous functions $\mathbb{R} \to \mathbb{R}$ which commute with the equivalence relation. (See here for more motivation.)

Dustin Clausen and Peter Scholze have a theory of condensed sets, which is a slightly different take on topology. For most cases, the behavior of the usual topology and the “condensed” one, align. However, for some quotient spaces, like $\mathbb{R} / \mathbb{Q}$, the usual quotient topology is indiscrete, so every map $f : \mathbb{R} / \mathbb{Q} \to \mathbb{R} / \mathbb{Q}$ is continuous. However, as a condensed set, more structure is preserved. Indeed, the “continuous” maps are exactly those coming from continuous functions $\mathbb{R} \to \mathbb{R}$ which commute with the equivalence relation. *[Edit: I was a bit careless here. The maps actually come from continuous multifunctions. See Arno's answer below.) (See here for more motivation.)

I’m trying to understand condensed sets better and how they relate to topology as it comes up in computable analysis, which I’m quite familiar with. I think the theory of p“representations" in computable analysis, is essentially the study of "countably generated condensed sets", and I want to know if anyone has worked this out, or thought about this?

If one only uses separable profinite sets in the definition of condensed set in place of profinite sets, let’s call those countably generated condensed sets. I conjecture the category of representations is the same as the category of countably generated condensed sets. Has anyone worked something like this out?

I’m trying to understand condensed sets better and how they relate to topology as it comes up in computable analysis, which I’m quite familiar with. I think the theory of “represented spaces" in computable analysis, is essentially the study of "countably generated condensed sets", and I want to know if anyone has worked this out, or thought about this?

If one only uses separable profinite sets in the definition of condensed set in place of profinite sets, let’s call those countably generated condensed sets. I conjecture the category of representations is the same as the category of countably generated condensed sets. Has anyone worked something like this out?

[Edit: This definition of countably generated condensed sets is definitely not what I'm looking for. Restricting the size of the profinite sets doesn't restrict the size of the set $X$.]

I’m trying to understand condensed sets better and how they relate to topology as it comes up in computable analysis, which I’m quite familiar with. I think the theory of “representations" in computable analysis, is essentially the study of "countably generated condensed sets", and I want to know if anyone has worked this out, or thought about this?

In computable analysis, a representation of a set $X$ is a partial, but surjective map $\rho : {\subseteq}2^\omega \to X$. (See sections 2 and 3 here.) There are four types of morphisms for the category of representations: (partial/total) (computable/continuous) maps. Let’s just consider total continuous maps even though usually in computable analysis one considers computable maps. Given two sets $X$ and $Y$ with representations $\rho : {\subseteq}2^\omega \to X$ and $\sigma : {\subseteq}2^\omega \to Y$, then a total function $f : X \to Y$ is $(\rho, \sigma)$-continuous if there is a corresponding partial total continuous function $f' : \text{dom}(\rho) \to \text{dom}(\sigma)$ such that $$\rho \circ f = f' \circ \sigma$$

I’m trying to understand condensed sets better and how they relate to topology as it comes up in computable analysis, which I’m quite familiar with. I think the theory of p“representations" in computable analysis, is essentially the study of "countably generated condensed sets", and I want to know if anyone has worked this out, or thought about this?

In computable analysis, a representation of a set $X$ is a partial, but surjective map $\rho : {\subseteq}2^\omega \to X$. (See sections 2 and 3 here.) There are four types of morphisms for the category of representations: (partial/total) (computable/continuous) maps. Let’s just consider total continuous maps even though usually in computable analysis one considers computable maps. Given two sets $X$ and $Y$ with representations $\rho : {\subseteq}2^\omega \to X$ and $\sigma : {\subseteq}2^\omega \to Y$, then a total function $f : X \to Y$ is $(\rho, \sigma)$-continuous if there is a corresponding continuous function $f' : \text{dom}(\rho) \to \text{dom}(\sigma)$ such that $$\rho \circ f = f' \circ \sigma$$