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