TITLE: What is the relation between a ''homotopy fiber bundle'' and a Serre fibration?
QUESTION [8 upvotes]: Since I got no responses to this question at Stack Exchange, please let me try my luck here.
Call a continuous map $\pi:E\to B$ between CW complexes a homotopy fiber bundle if for any $x$ in the image of $\pi$, there is an open neighbourhood $U\subset B$ of $x$ and homotopy equivalence $\pi^{-1}(U)→U\times F$ over $U$.
I don't know if this has a different name in the literature or even if it is reasonable. Replacing ''homotopy equivalence'' by ''homeomorphism'' should be the definition of an ordinary fiber bundle.

How relates a ''homotopy fiber bundle'' to the notion of a Serre fibration?

At least both properties imply that the fibers over connected components are all weakly homotopy equivalent.

REPLY [10 votes]: There is a nice theorem in geometric topology that a Serre fibration between
actual CW complexes (not just homotopy types thereof) is in fact a Hurewicz fibration.
It's in a paper by Steinberger and West (Covering homotopy properties of maps between
CW complexes or ANRs, Proc AMS92(1984), 573-577), with a correction in a paper by
R. Cauty (Sur les ouverts des CW-complexes et les fibr\'es de Serre, Colloq. Math. 63(1992), 1-7).
Then the rest of your answer applies. But of course the proof of the cited theorem proceeds
by checking homotopical local triviality and then quoting the local to global characterization
of Hurewicz fibrations.