nLab
covering dimension

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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)