A fibration, $p\colon E\to B$ and a continouos map $\pi \colon F\to B$ , and a homotopy equivalence between $E$ and $F$ which respects the fibres gives $\pi \colon F\to B$ the structure of a fibration. Is there an example of a fibration which does not arise from this construction with $p\colon E\to B$ being a fibre bundle?
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https://math.stackexchange.com/questions/1380462/gap-between-fibration-and-fiber-bundle?rq=1 – Moishe Kohan Feb 11 '18 at 13:33