nLab
stacked cover

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 $A,B$ be topological spaces and $𝒰$ a numerable cover of $A$. Then a cover of the product space $A\times B$ is called a stacked cover of $A\times B$ over $𝒰$ – denoted $𝒰\times 𝒮$ – , if there exists a function $𝒮$ – called the stacking function – which assignes to each set $U\in 𝒰$ a cover $𝒮U$ of $B$, such that $𝒰\times 𝒮$ consists of all the sets $U\times V$ with $V\in 𝒮U$.

Properties

General

Proposition

A stacked cover is itself a numerable cover.

Stacked covers of products with the interval

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

Proposition

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

$$\{U_i\times [\frac{k-1}{r_i},\frac{k+1}{r_i}]\mid r_i,k\in \mathbb{N},1\le k\le r_i-1\}.$$\{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)