nLab
stacked cover

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 U𝒰U \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 V𝒮UV \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×[k1r i,k+1r i]r i,k,1kr i1}. \{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)