TITLE: Is there a category of topological-like spaces that forms a topos?
QUESTION [5 upvotes]: The category of convergence spaces generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a topos?

REPLY [2 votes]: Let me point out a couple of other possibilities, they may come handy in certain situations. The realizaibility topos $RT(P(\omega))$ over Scott's graph model $P(\omega)$ contains countably based spaces as a full subcategory. The inclusion preserves countable limits, countable coproducts, and those exponentials which happen to exist in countably based spaces. This can be generalized to spaces of arbitrary weight by using a larger graph model $P(\kappa)$. But as Todd points out, such a topos contains a lot of non-topological junk.
If you do not insist on a topos, and are willing to consider slightly less than a topos, then the exact completion of topological spaces is a good candidate. The objects are topological spaces equipped with a (formal) equivalence relation, so they feel like spaces still. Not too much junk there. For this sort of thing, see

Carboni, A., Rosolini, G. Locally cartesian closed exact completions, J.Pure Appl. Alg., 154, 2000.

A naive way to get spaces inside a topos is to simply consider (pre)sheaves on all spaces, which of course runs against size restrictions. Nevertheless, a useful picture emerges, see

Rosolini, G., Equilogical spaces and filter spaces, Rend. Circ. Mat. Palermo, 64, 2000