Let $X$ be a compact Hausdorff space, and let $\pi: E \to X$ be a real or complex vector bundle over $X$ of rank $r$. Then there exist finitely many local trivializations of $E$, i.e. there exists an open covering $(U_j)_{j=1}^k$ and homeomorphisms $h_j : \pi^{-1}(U_j) \to U_j \times \mathbb{K}^r$ such that $\pi(h_j^{-1}(x,v)) = x$ for $j=1, \ldots, k$.
Is there a name for the least such $k$, i.e. the least number of local trivializations required for the vector bundle? Also, are there some references looking into this concept?