Are there topoi in topology

The straight answer is no. Topoi are too set-like to occur as categories of sets with topological structure. However, if A is a category of sets with structure, and if A has enough substructures, then A has a full and dense embedding into a complete quasitopos of sets with structure. There is a minimal embedding of this type; it embeds e.g. topological spaces into the quasitopos of Choquet spaces. Quasitopoy are still very set-like. They are cartesian closed, and all colimits in a quasitopos are preserved by pullbacks. Thus quasitopoi are in a sense ultra-convenient categories for topologists. Quasitopoi inherit many properties from topoi. For example, the theory of geometric morphisms of topoi remains valid, almost without changes, for quasitopoi.