Numerable fiber bundles are defined by Dold (DOLD 1962 – Partitions of Unity in theory of Fibrations) as a generalization of fiber bundles over a paracompact space : the trivialization cover of the base admit a subordinate partition of unity (locally finite). He proves in this paper that almost all important theorems for fiber bundles over paracompact spaces are also valid for numerable bundles.
But is it really an interesting generalization ? Are there examples of "natural" or "useful" numerable fiber bundles that are not paracompact?
Best Answer
In algebraic topology, it is often more convenient to know that a map is a fibration (has the homotopy lifting property with respect to all spaces) than a fibre bundle, because then calculational tools such as long exact sequences of homotopy groups and Serre spectral sequences of (co)homology groups become available.
It is easy to cook up examples of fibrations which are not fibre bundles (the projection of a $2$-simplex onto one of its edges being the easiest example I know). It is somewhat harder to find examples of fibre bundles which are not fibrations, but they do exist; see here and here.
Numerability is precisely the extra condition on fibre bundles which makes them into fibrations. Of course this means that any fibre bundle over a paracompact base is a fibration.
The homotopy lifting property is used extensively when proving the homotopy classification of principal $G$-bundles, i.e. that isomorphism classes of principal $G$-bundles a given base space $B$ are in one-to-one correspondence with homotopy classes $[B,BG]$. To obtain such a result for arbitrary base spaces $B$ you had better therefore restrict to numerable bundles.
This doesn't really answer your question, in that I haven't given you a natural example of a numerable bundle over a non-paracompact base. But hopefully it indicates why numerable bundles are a useful concept in homotopy theory.