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
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 priori a locally compact topological group is a topological group $G$ whose underlying topological space is locally compact.
Typically it is also assumed that $G$ is Hausdorff. (Notice that if not, then $G/\overline{\{1\}}$ is Hausdorff.).
One often says just “locally compact group”.
We take here locally compact groups $G$ to be also Hausdorff.
Locally compact topological groups are the standard object of study in classical abstract harmonic analysis. The crucial properties of locally compact groups is that they posses a left (right) Haar measure $\rho$ and that $L^1(\rho)$ has a structure of a Banach $*$-algebra.
A left (right) Haar measure on a locally compact topological group is a nonzero Radon measure which is invariant under the left (right) multiplications by elements in the group. A topological subgroup $H$ of a locally compact topological group $G$ is itself locally compact (in induced topology) iff it is closed in $G$.
Again taking locally compact groups $G$ to be Hausdorff, such are complete both with respect to their left uniformity and their right uniformity. For if $\{x_\alpha\}$ is a Cauchy net in $G$ and $U$ is a compact neighborhood of the identity $e$, then there is $\alpha$ so large that $x_\beta x_\alpha^{-1} \in U$ for all $\beta \geq \alpha$. Those elements converge to a point $x \in U$ since $U$ is compact, and the original net converges to $x \cdot x_\alpha$. A similar argument is used for the right uniformity.