nLab
numerable open cover

Idea

A numerable open cover is an open cover of a topological space that admits a subordinate partition of unity.

Definition

Let $\{U_i\}$ be an open cover of the topological space $X$ (actually Dold doesn’t always require open, see discussion below). It is said to be numerable if there is a collection of functions ${\varphi }_i:X\to [0,1]$ such that

• $\overline{\mathrm{supp}({\varphi }_i)}\subset U_i$,
• at each point $x\in X$, only finitely many of the ${\varphi }_i$ are non-zero,
• ${\sum }_i{\varphi }_i(x)\equiv 1\forall x\in X$.

The open cover ${\varphi }_i^{-1}(0,1]$ is then a locally finite cover that refines $\{U_i\}$. The functions $\{{\varphi }_i\}$ are a partition of unity.

Numerable open covers form a site called the numerable site. Many classical theorems concerning bundles are stated for the numerable site. For example, the classifying space $ℬG$ actually classifies bundles which trivialise over a numerable cover. (References? Dold for Milnor’s classifying space, and tom Dieck I think for Segal’s) These are called numerable bundles?.

For paracompact spaces, numerable covers are cofinal in open covers, so that the numerable site is equivalent to the open cover site. (This needs some more and better saying -David R)

There is also some result by Bourbaki that I have to look up that numerable covers are cofinal in locally finite covers of normal spaces.

References

• A. Dold, Partitions of unity in the theory of fibrations

Another one by Dold, will look it up - it talks about stacked covers in the appendix: these are useful for ‘decomposing’ numerable covers of products to a sort of parameterised version depending on a numerable cover of the first factor. This is important in looking at concordance of numerable bundles.