see also algebraic topology, functional analysis and homotopy theory
topological space (see also locale)
fiber space, space attachment
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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
Theorems
A point of a topological space is called focal (Freyd-Scedrov) if its only open neighbourhood is the entire space.
The closed point $\bot$ of Sierpinski space $\mathbf{2}$ is a focal point.
The vertex $v$ of a Sierpinski cone (or scone) $s(X)$ on a space $X$, given by a pushout in $Top$
is a focal point. This construction is in fact the same as generically adding a focal point to $X$.
The prime spectrum of a ring $A$ has a focal point iff $A$ is a local ring. In this case, the focal point is given by the unique maximal ideal of $A$.
The category of sheaves over (the site of open subsets) of a topological space with focal point is a local topos.
Every topos has a free “completion” to a “focal space”, given by its Freyd cover.
A locale $X$ is called local if in any covering of $X$ by opens $U_i$, at least one $U_i$ is $X$.
The locale associated to an sober space is local in this sense if and only if the space possesses a focal point (see Johnstone, discussion preceding lemma C1.5.6). Locale theoretically, this point is then given by the frame homomorphism
where $\Omega$ is the frame of opens of the point.