∞-Lie theory (higher geometry)
see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
A Lie groupoid $(C_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} C_0)$ is proper if its underlying topological groupoid is a proper topological groupoid, hence if
is a proper map.
So in particular the automorphism group of any object in a proper Lie groupoid is a compact Lie group. In this sense proper Lie groupoids generalize compact Lie groups.