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 $\mathcal{U}$ 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 $\mathcal{U}$ – denoted $\mathcal{U} \times \mathcal{S}$ – , if there exists a function $\mathcal{S}$ – called the stacking function – which assignes to each set $U \in \mathcal{U}$ a cover $\mathcal{S}U$ of $B$, such that $\mathcal{U} \times\mathcal{S}$ consists of all the sets $U \times V$ with $V \in \mathcal{S}U$.

Properties

General

Proposition

A stacked cover is itself a numerable cover.

Stacked covers of products with the interval

In this section we let $B = [0,1]$ 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 $\mathcal{W}$ a numerable cover of $A \times [0,1]$ there exists a refinement of $\mathcal{W}$ to a stacked cover $\mathcal{U} \times \mathcal{S}$ of $A \times [0,1]$ of the form

$$\{U_i \times [\frac{k-1}{r_i}, \frac{k+1}{r_i}] | r_i,k \in \mathbb{N}, 1 \leq k \leq r_i-1\} \,.$$

References

Section A.2.17 of

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