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 finite topological space is a topological space whose underlying set is a finite set.
Every finite topological space is an Alexandroff space.
I.e. finite topological spaces are equivalent to finite preordered sets, by the specialisation order.
Finite topological spaces have the same weak homotopy types as finite simplicial complexes / finite CW-complexes.
This is due to (McCord 67).
If $\mathbf{2}$ is Sierpinski space (two points $0$, $1$ and three opens $\emptyset$, $\{1\}$, and $\{0, 1\}$), then the continuous map $I = [0, 1] \to \mathbf{2}$ taking $0$ to $0$ and $t \gt 0$ to $1$ is a weak homotopy equivalence^{1}.
For any finite topological space $X$ with specialization order $\mathcal{O}(X)$, the topological interval map $I \to \mathbf{2}$ induces a weak homotopy equivalence $B\mathcal{O}(X) \to X$:
(where we implicitly identify $\Delta^{op}$ with the category $Int$ of finite intervals with distinct top and bottom). The isomorphism on the right says that any finite topological space can be constructed by gluing together copies of Sierpinski space, in exactly the same way that any preorder can be constructed by gluing together copies of the preorder $\{0 \leq 1\}$.
On the other hand, any finite simplicial complex $X$ is homotopy equivalent to its barycentric subdivision, which is the geometric realization of the poset of simplices ordered by inclusion. Thus finite posets model the weak homotopy types of finite simplicial complexes.
A survey is in
published as
The original results by McCord are in
Michael C. McCord, Singular homology groups and homotopy groups of finite topological spaces , Duke Math. J. 33 (1966), 465-474. (EUCLID)
Michael C. McCord, Homotopy type comparison of a space with complexes associated with its open covers . Proc. Amer. Math. Soc. 18 (1967), 705-708, copy
Generalization to ringed finite spaces is discussed in
Fernando Sancho de Salas, Ringed Finite Spaces (arXiv:1409.4574)
Fernando Sancho de Salas, Finite Spaces and Schemes (arXiv:1602.02393)
and aspects of their homotopy theory is discussed in
Any topological meet-semilattice $L$ with a bottom element $\bot$, for which there exists a continuous path $\alpha \colon I \to L$ connecting $\bot$ to the top element $\top$, is in fact contractible. The contracting homotopy is given by the composite $I \times L \stackrel{\alpha \times 1}{\to} L \times L \stackrel{\wedge}{\to} L$. ↩