Locality of Serre fibrations

Theorem 2.3.13 (Locality for Serre fibrations, tom Dieck (2008), Theorem 6.3.3). Let \(p\colon E \to B\) be a continuous map, and assume that there exists an open cover \(\{U_i\}_{i \in I}\) such that the restriction \(p\vert _{U_i} \colon p^{-1}(U_i) \to U_i\) is a Serre fibration for each \(i\in I\). Then \(p\) is itself a Serre fibration.

In particular, locally trivial fiber bundles are Serre fibrations.