Johnstone’s topological topos (specifically, the one described in the eponymous paper referenced below) is the topos of canonical sheaves$Sh_J(\Sigma)$ on $\Sigma$. The functor

$Top \to Set^{\Sigma^{op}}: X \mapsto Top(-, X)$

is faithful and factors through $Sh_J(\Sigma)$, and its restriction to the category of sequential spaces is full.

The category of subsequential spaces can also be found as a full subcategory of this topos (in fact, it consists of the separated objects for a Lawvere-Tierney topology). A general object of the topos can be thought of as like a subsequential space, but such that a given sequence can converge to a given point in “more than one way.”

Importantly, following an idea by Joyal, the topological topos $\mathcal{E}:=Sh_J(\Sigma)$ allows one to represent the geometric realization functor $sSet \to \mathcal{E}$ as the inverse image of a geometric morphism from $\mathcal{E}$ to the topos of simplicial sets. As Johnstone points out, this approach fails for the big topos on $Top$ and also for Lawvere’s topos for continuum mechanics.

Reference

Peter Johnstone, On a topological topos, Proc. London Math. Soc. (3) 38 (1979) 237–271. (PDF)