nLab
Johnstone's topological topos

Definition

Let Top be the category of topological spaces, and let $\Sigma$ be the full subcategory whose only two objects are a one-point space and ${\mathbb{N}}^{+}$, the one-point compactification of the discrete space of natural numbers. Let $J$ be the canonical Grothendieck topology on $\Sigma$.

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

is faithful and factors through ${\mathrm{Sh}}_J(\Sigma )$, and its restriction to the category of sequential spaces is full.

Reference

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