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 topological space $X$ is said to be locally contractible if it has a basis of open subsets that consists of contractible topological spaces $U \hookrightarrow X$.
Sometimes one requires just that the inclusions $U \to X$ are null-homotopic map?s. This might be called semi-locally contractible.
One could also consider a basis of open sets such that the opens $U$ have (just) trivial homotopy groups, but this does not seem to crop up in practice.
A locale $X$ is locally contractible if, viewing a locale as a $(0,1)$-topos and hence a (very special kind of) $(\infty,1)$-topos, it is locally ∞-connected.
Is this right? Do these two definitions correspond in that a sober space or topological locale is locally contractible as a topological space iff it's locally contractible as a locale? —Toby
For $X$ a locally contractible topological space, the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(X)$ is a locally ∞-connected (∞,1)-topos.
This is discussed at locally ∞-connected (∞,1)-site.
If one considers fundamental ∞-groupoids, the inclusion $U \to X$ being null-homotopic is equivalent to the induced (∞,1)-functor $\Pi(U) \to \Pi(X)$ being naturally isomorphic to the trivial functor sending everything to a single point.
David Roberts: The following may be straightforwardly obvious, but I have couched it as a conjecture, because I haven’t seen it in print.
If the space $X$ is semi-locally contractible then every locally constant $n$-stack on the site of open sets of $X$ is locally trivial.
See also locally ∞-connected (∞,1)-topos. There a converse to this conjecture is stated:
Let $C$ be a site coming from a coverage such that constant (∞,1)-presheaves satisfy descent over objects of $C$ with respect to the generating covering families. Then the (∞,1)-category of (∞,1)-sheaves $\mathbf{H} = Sh_{(\infty,1)}(C)$ is a locally ∞-connected (∞,1)-topos.