nLab covering dimension

Context

Topology

topology

algebraic topology

Contents

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 disting $V_i$ is empty

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

Properties

Theorem

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

This is HTT, theorem 7.2.3.6.

Created on January 12, 2011 13:19:47 by Urs Schreiber (89.204.137.103)