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 filtered space $X_*$ is called a connected filtered space if it satisfies:
$(\phi)_0$: The function $\pi_0X_0 \to \pi_0X_r$ induced by inclusion is surjective for all $r \gt 0$; and,
for all $i \geq 1$, $(\phi_i): \pi_i(X_r,X_i,v)=0$ for all $r \gt i$ and $v \in X_0$.
Another equivalent form is:
$(\phi_0')$: The function $\pi_0X_s \to \pi_0X_r$ induced by inclusion is surjective for all $0=s \lt r$ and bijective for all $1 \leq s \leq r$; and,
for all $i \geq 1$, $(\phi_i'): \, \pi_j(X_r,X_i,v)=0$ for all $v \in X_0$ and all $j,r$ such that $1 \leq j \leq i \lt r$.
Examples of connected filtered spaces are:
The skeletal filtration of a CW-complex.
The word length filtration of the James construction for a space with base point such that $\{x\} \to X$ is a closed cofibration.
The filtration $(B C)_*$ of the classifying space of a crossed complex, filtered using skeleta of $C$.
This condition occurs in the higher homotopy van Kampen theorem for crossed complexes.