nLab
covering dimension

Definition

Definition

A paracompact topological space $X$ has covering dimension $\le n\in \mathbb{N}$ 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

$$V_{i_0}\cap \cdots \cap V_{i_{n+1}}=\varnothing .$$V_{i_0} \cap \cdots \cap V_{i_{n+1}} = \emptyset \,.

Properties

Theorem

If the paracompact topological space $X$ has covering dimension $\le n$, then the (∞,1)-category of (∞,1)-sheaves ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(X):={\mathrm{Sh}}_{(\mathrm{\infty },11)}(\mathrm{Op}(X))$ is an (∞,1)-topos of homotopy dimension $\le n$.

This is HTT, theorem 7.2.3.6.

Related concepts

• dimension

  • homotopy dimension

  • cohomology dimension

  • covering dimension

  • Heyting dimension