I'm currently studying bundles from Husemoller "Fibre Bundles" and I'd like to understand the definition of partition of unity given in the book :
I think for the first time there's a reference (Theorem $4.3$,p.$29$) where the author says
"Let $(\eta_i)_{i \in I}$ be an envelope of unity subordinate to the open covering $(U_i)_{i \in I}$, that is, the support of $\eta_i \subset U_i$ and $\max_{i \in I}\eta_i(b) = 1$".
I think this is a slightly different notion of partition of unity given here and I'd like to understand in this context if the latter follows from having a Partition_of_unity. For example the "envelope" doesn't seem to ask the condition on the closure on the support.
Morover, p.$31$ Proposition $5.4$ Husemoller takes a partition of unity $(\eta_i)_{i \in I}$ using only paracompactness of $B$. This seems in contradiction with p.$50$ where "It is a standard result that a Hausdorff space $B$ is paracompact if and only if each open covering is numerable". My question here is : In Proposition $5.4$ there are some properties on the space (i.e. $B$ Hausdorff) in order to have a partition of unity?
Any help would be appreciated.
Edit : In particular I'd like to whether there's a proof of the existence of an "envelope of unity subordinate to the open covering" and its dependance with the existance of a partition of unity and hyphotesis on the spaces.