CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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.