nLab
covering dimension

Contents

Definition

Definition

A paracompact topological space XX has covering dimension n\leq n \in \mathbb{N} if for every open cover {U iX}\{U_i \to X\} there exists an open refinement {V iX}\{V_i \to X\}, such that each (n+1)(n+1)-fold intersection of pairwise disting V iV_i is empty

V i 0V i n+1=. V_{i_0} \cap \cdots \cap V_{i_{n+1}} = \emptyset \,.

Properties

Theorem

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