topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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
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
continuous metric space valued function on compact metric space is uniformly continuous
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
Analysis Theorems
A locally compact topological group $G$ is called almost connected if the underlying topological space of the quotient topological group $G/G_0$ (of $G$ by the connected component of the neutral element, also called the identity component) is compact.
See for instance (Hofmann-Morris, def. 4.24). We remark that since the identity component $G_0$ is closed, the identity in $G/G_0$ is a closed point. It follows that $G/G_0$ is $T_1$ and therefore, because it is a uniform space, $T_{3 \frac1{2}}$ (a Tychonoff space; see uniform space for details). In particular, $G/G_0$ is compact Hausdorff.
Every compact and every connected topological group is almost connected.
Also every quotient of an almost connected group is almost connected.
Textbooks with relevant material include
M. Stroppel, Locally compact groups, European Math. Soc., (2006)
Karl Hofmann Sidney Morris, The Lie theory of connected pro-Lie groups, Tracts in Mathematics 2, European Mathematical Society, (2000)
Original articles include