Let $p:E\rightarrow B$ be a fibration and $E_0\subset E$ a subset.
When is the restriction $p_0=p|:E_0\rightarrow B$ also a fibration?
It seems like there should be some simple conditions which guarantee this, but I couldn't find any. For concreteness we may assume that $p$ is a surjective Hurewicz fibration (i.e. it has the covering homotopy property) over a connected base $B$, and that $p_0$ is also surjective. I'm also willing to assume that all spaces are nice (even CW). I would still be interested in knowing some of the more general results, however.