CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The category $TopMfd$ is that whose
objects are topological manifolds;
morphisms are continuous functions between these.
The category $TopMfd$ becomes a large site equipped with the open cover coverage. This is a small-generated site: a dense sub-site is CartSp${}_{top}$.
The hypercompleted (∞,1)-sheaf (∞,1)-topos over $TopMfd$ is that of Euclidean-topological ∞-groupoids.
See there for details.
TopMfd