Hurewicz fibration

Remarks

Example: for spaces $B$ and $F$, the projection $p \colon B \times F \to B$ is a Hurewicz fibration. Given $H$ and $\tilde{h}_0 = (H(-, 0), g)$, the lift $\tilde{H}(x, t) = (H(x, t), g(x))$ works.

Non-example: the inclusion $p \colon \set{0} \to I$ is not a fibration. Take $X$ a point, $H(t) = t$ and $\tilde{h}_0 = 0$: a lift would satisfy $p(\tilde{H}(1)) = 1$, but $1$ is not in the image of $p$.

Covering maps and fibre bundles over paracompact Hausdorff bases are Hurewicz fibrations. If $p$ is a Hurewicz fibration and $B$ is path-connected, all fibres $p^{-1}(b)$ are homotopy equivalent. Every Hurewicz fibration is a Serre fibration, where the lifting property is only required for $X$ a cube $I^n$ or a CW complex, but not conversely.