nLab
stacked cover

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

Idea

A stacked cover is a cover of a topological space which is indexed by a cover of another topological space, such that the product cover is a cover of the product space.

Definition

Definition

Let A,BA,B be topological spaces and 𝒰\mathcal{U} a numerable cover of AA. Then a cover of the product space A×BA \times B is called a stacked cover of A×BA\times B over 𝒰\mathcal{U} – denoted 𝒰×𝒮\mathcal{U} \times \mathcal{S} – , if there exists a function 𝒮\mathcal{S} – called the stacking function – which assignes to each set U∈𝒰U \in \mathcal{U} a cover 𝒮U\mathcal{S}U of BB, such that 𝒰×𝒮\mathcal{U} \times\mathcal{S} consists of all the sets U×VU \times V with V∈𝒮UV \in \mathcal{S}U.

Properties

General

Proposition

A stacked cover is itself a numerable cover.

Stacked covers of products with the interval

In this section we let B=[0,1]B = [0,1] the standard interval and consider properties of stacked covers of spaces of the form A×[0,1]A \times [0,1].

Proposition

For AA a topological space and 𝒲\mathcal{W} a numerable cover of A×[0,1]A \times [0,1] there exists a refinement of 𝒲\mathcal{W} to a stacked cover 𝒰×𝒮\mathcal{U} \times \mathcal{S} of A×[0,1]A \times [0,1] of the form

{U i×[k−1r i,k+1r i]|r i,k∈ℕ,1≤k≤r i−1}. \{U_i \times [\frac{k-1}{r_i}, \frac{k+1}{r_i}] | r_i,k \in \mathbb{N}, 1 \leq k \leq r_i-1\} \,.

References

Section A.2.17 of

  • Albrecht Dold, Lectures on algebraic topology , Spring Verlag (1980)

Revised on August 17, 2010 17:32:21 by Urs Schreiber (131.211.232.139)