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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 $\pi(x)$ 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.

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A Serre fibration is defined by a lifting property against maps $X \to X \times [0;1]$ for cubes $X$. There is some kind of homotopy version of this, which is called the "weak covering homotopy property" (WHCP). Maps with the WHCP for all spaces are called Dold fibrations and these turn out to be "locally homotopy trivial maps", such as your homotopy fiber bundle.

Check out for more detailled definitions.

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