The open cover is then a locally finite cover that refines . The functions 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 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)
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.