Remark 2

Let $(E, p)$ be a B-space and let $(E_i)_{i\in I}$ be a partition of E consisting of open sets.

If each of the B-spaces $(E_i, p|E_i)$ is a trivializable fibre space, E is a trivializable fibred B-space. In fact, for each $i\in I$, let $F_i$ be a topological space such that the B-spaces $(E_i, p|E_i)$ and $(B\times F_i$, pr$_1)$ are isomorphic. Then the B-space $(E, p)$ is isomorphic to $(B\times F$, pr$_1)$, where F is the topological space sum of the family $(F_i)_{i\in I}$.

Suppose the set I finite. If each of the B-spaces $(E_i, p|E_i)$ is a locally trivial fibred B-space, the same is true of the B-space E. In fact, since the set I is finite, every point of B possesses a neighbourhood U above which the fibred B-spaces $E_i$ are all trivializable. By what precedes, the U-space $(\overset{-1}{p}(U), p_U)$ is then a trivializable fibre space.