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 paracompact topological space $X$ has covering dimension $\leq n \in \mathbb{N}$ if for every open cover $\{U_i \to X\}$ there exists an open refinement $\{V_i \to X\}$, such that each $(n+1)$-fold intersection of pairwise disting $V_i$ is empty
If the paracompact topological space $X$ has covering dimension $\leq n$, then the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(X) := Sh_{(\infty,11)}(Op(X))$ is an (∞,1)-topos of homotopy dimension $\leq n$.
This is HTT, theorem 7.2.3.6.
covering dimension