nLab covering dimension

Definition

Definition

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 distinct $V_i$ is empty

$$V_{i_1} \cap \cdots \cap V_{i_{n+1}} = \emptyset \,.$$

Properties

Theorem

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

This is HTT, theorem 7.2.3.6.

Remark

For separable metric spaces the notion of covering dimension is particularly well-behaved. See there.

Related concepts

notion of dimension

• homotopy dimension

• cohomology dimension (virtual)

• covering dimension

• Heyting dimension

• dimension of a manifold

• dimension of a cell complex

• dimension of a separable metric space

• small inductive dimension of a regular space

• large inductive dimension of a normal space