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I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts: The category of compact Hausdorf ...

As has already been remarked, $\mathbf{CHaus}$ is monadic over $\mathbf{Set}$. It is also a pretopos, meaning roughly that it has the finitary properties of $\mathbf{Set}$. Another more symmetrical ...

We can describe $\mathbf{CHaus}$ with a universal property inside the $2$-category of all cocomplete categories and cocontinuous functors. Namely, $\mathbf{CHaus}$ is the universal cocomplete category ...

David Roberts has rubbed the magic lamp and the genie appears! Even though the notion of overtness does depend on the strength of the ambient logic, I believe the question here is with the notion of ...

The spectrum of a commutative ring, defined as the classifying locale of its prime filters, is overt if and only if any element is nilpotent or not nilpotent (Proposition 12.51 in these notes of mine)....

To conjure up a non-overt space we must change slightly the definition of topology, since even inuitionistically every space is overt, so long as every union of opens is open. Let $\Sigma$ be the ...

In computable analysis, the typical approach to metric spaces is that of a computable metric space. If we assume that there already is some external concept of the metric space we want to handle, we ...

The following (somewhat strong) property seems to be sufficient: For any open sets $U,V \subseteq \mathbb{R}$, if $\forall x \in \mathbb{R}.\; x \in U \vee x \in V$, then either $S \subseteq U$ or $S ...

In Lifschitz realizability $\mathbf{LLPO}$ holds but the intermediate value theorem fails. The fact that $\mathbf{LLPO}$ holds is a standard property of Lifschitz realizability. The analytic version ...