Can someone provide me a proof or a reference of the fact that fibres of a Hurewicz fibration $E \to B $ are homotopic equivalent for a path-connected space $B$. This question has been asked before here, but I am not convinced with the answer. For instance uniqueness of lifting is assumed which is not true for fibrations.
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1If you delete the uniqueness in the link, doesn't the argument still go through? – Randall Dec 09 '22 at 03:15
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What precisely are your doubts with the linked answer? – Paul Frost Dec 09 '22 at 11:28
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1One of the issues would be that the map $\tilde{h}$ won't be well defined since I have to make a choice for the lift of the path. – Reznick Dec 09 '22 at 20:00
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But $\overline{h}$ is the desired homotopy, and it need not be unique to get the result. – Randall Dec 09 '22 at 21:23