In Peter Johnstone's 1979 paper On a topological topos, he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}_\infty$, the one-point compactification of the natural numbers, as a convenient big topos whose objects are a sort of space. He showed that this topos (now sometimes known as Johnstone's topological topos) has the following nice properties:

- It is a local topos, i.e. its global sections functor (whose left adjoint constructs "discrete spaces") has a fully faithful right adjoint, which constructs "indiscrete spaces". (But it is not locally connected, hence not "cohesive".)
- It contains the category of sequential topological spaces as a full subcategory closed under limits (and indeed reflective) --- although sequential spaces are
*coreflective*in all topological spaces, so limits in the former don't coincide with limits in the latter. - The embedding of sequential spaces preserves colimits arising from open covers, closed covers, and quotients of sequentially closed equivalence relations. In particular, therefore, it preserves the construction of CW-complexes.
- The internally-constructed real numbers object coincides with the usual (sequential) space of real numbers with its usual topology.
- Geometric realization is the inverse image part of a geometric morphism from this topos to the topos of simplicial sets.

Much more recently, Barwick and Haine have introduced the topos of pyknotic sets with, it seems to me, a similar motivation. (The Scholze-Clausen category of condensed sets is related, but not a topos, so for purposes of this question I'm not as interested in it.) In some ways, pyknotic sets feel to me like an extension of Johnstone's topological topos to non-sequential notions of convergence. By definition, it consists of sheaves on a category of compact Hausdorff topological spaces, which includes $\mathbb{N}_\infty$. And the paper Pyknotic objects, I proves that the topos of pyknotic sets has analogues of properties 1 and 2 above, and partly 3:

Pyknotic sets is a local topos.

It contains the category of compactly generated spaces as a full subcategory closed under limits, while compactly generated spaces are coreflective in all topological spaces.

This embedding preserves sequential colimits of compactly generated spaces whose colimit is $T_1$.

Thus, I wonder whether the topos of pyknotic sets shares any of the other nice properties of Johnstone's topological topos. Namely:

Does the embedding of compactly generated spaces preserve any more colimits, such as those arising from open or closed covers? In particular, does it preserve the construction of CW-complexes?

Does the real numbers object in pyknotic sets coincide with the usual (compactly generated) space of real numbers with its usual topology?

Is geometric realization the inverse image part of a geometric morphism from pyknotic sets to simplicial sets?

2more comments