For $p: (E, e_0) \to (B, b_0)$ a Hurewicz fibration, is the inclusion $p^{-1}(b_0) \to E$ a Hurewicz fibration as well?

Question

As $p$ is a Hurewicz fibration, its homotopy fiber $hofib_{b_0}(p)$ is homotopy equivalent to $p^{-1}(b_0)$, but does this also mean that the map $p^{-1}(b_0) \to E$ is a Hurewicz fibration as well?

Answer 1

No. For example, the identity $id : (\mathbb R, 0) \to (\mathbb R, 0)$ is a Hurewicz fibration, but the inclusion $\{0\} = id^{-1}(0) \to \mathbb R$ is not. See Surjectivity of Hurewicz fibrations: For a Hurewicz fibration $p : E \to B$ the set $p(E)$ is a union of path components of $B$.