I am taking a first course in differential geometry, and we have looked at vector bundles and principle G bundles (I suppose that the following will hold for fibre bundles in general where the base is a manifold). My query concerns a point in several of the proof we have looked at, namely that we can find an open cover of the base manifold $B$, say $\{U_\alpha \}$, where the $U_\alpha \subset B$ are both coordinate charts and trivialising neighborhoods.
By trivialising neighborhood, I mean that there is a diffeomorphism $\Phi_\alpha$ such that $\Phi_\alpha : \pi^{-1}(U_\alpha) \rightarrow U_\alpha x V$ with $\pi : E \rightarrow B$ the map in the vector bundle and $V$ the typical fibre.
Naturally, it is nice if we have this is problems because then we can use local coordinates along with a chosen basis on the typical fibre, to have local coordinates on open sets of the total space $E$. We have used this assumption often when treating connections. Another case is in proving the existence of a smooth positive definite inner product on the total space E. We use this along with partition of unity, which requires the open neighborhoods that are both charts and trivialising neighborhoods to cover B.
EDIT:
I think in fact this is trivial. By virtual of being a fibre bundle, for every $p\in B$ we have an open neighborhood $p \in U \subset B$ and a corresponding local trivialisation $\phi _U$. Likewise, since $B$ is a manifold then we have an open cover of coordinate charts $V_\alpha$. Since topology is closed under finite intersection, we need simply take the intersection $U_p \cap V_\alpha$ where $p \in V_\alpha$ and take the union of these for all $p \in B$. Now we have our open cover of $B$ consisting of coordinate charts that are trivialising neihborhoods.