Example 6.73.
Let \(X\) be a paracompact Hausdorff space. We say that \(X\) has covering dimension \(\leq d\) if every open covering is refined by one with empty \((d+2)\)-fold intersections. Then
\[\dim \Shv(X) \leq d
\quad\Longleftrightarrow\quad
\text{$X$ has covering dimension $\leq d$}.\]
This applies, for example, to every CW-complex of dimension \(\leq d\). See [Lurie 2009, Theorem 7.2.3.6 and Corollary 7.2.3.7].
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.