higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
Kolmogorov space, Hausdorff space, regular space, normal space
connected space, locally connected space, contractible space, locally contractible space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
compact spaces equivalently have converging subnet of every net
Theorems
A topological groupoid is an internal groupoid in the category Top.
So this is a groupoid with a topological space of objects and one of morphisms, and all structure maps (source, target, identity, composition, inverse) are continuous maps.
A topological groupoid $C$ is called an open topological groupoid if the source map $s : Mor C \to Obj C$ is an open map.
It is called an étale groupoid if in addition $s$ is a local homeomorphism.