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Why can't we require in the definition of a trivial fibration that the map has the right lifting property with respect to each inclusion of a closed subset into a (hereditary) normal space, or perhaps with respect to each $C$-embedded subset ?

What are natural examples of trivial Hurewicz between "nice" spaces which are not fibrations in this stronger sense ?

I suppose it might be harder to give examples of trivial fibrations in this stronger sense. But I am mostly interested to see some examples and counterexamples:

i. a closed inclusion $i:A\subset B$ into a (better if hereditary) normal space $B$ and a trivial Hurewicz fibration $f$ of "nice" spaces such that $i$ does not lift with respect to $f$

ii. a trivial Hurewicz fibration $f$ which does lift with respect to all closed inclusions into (hereditary) normal spaces

The only result I know is the obvious remark that the map contracting a retract of $\mathbb{R}^n$ to a point, is a trivial fibration in this sense. I suppose these questions should have been discussed somewhere but I was not able to find anything that does not assume that it is a closed inclusion into a paracompact space of finite Lebesgue dimension, or the fibration is a map to a single point.

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  • $\begingroup$ Regarding the first paragraph: the answer to a question of the form "why can't we change the following definition" will depend on what you want to use the definition for. One use for trivial Hurewicz fibrations is as part of the Strom model structure. So one question to ponder is whether the Strom model structure could be modified (presumably the idea would be to keep the same weak equiavalences?) to use this notion of trivial fibration. There would be more cofibrations... $\endgroup$
    – Tim Campion ♦
    Jan 19, 2022 at 16:42
  • $\begingroup$ Indeed. A model structure would be nice, and (sort of) yes, the idea would be to keep the same weak equivalences/fibrations between "very nice" spaces, or even weaker, a "very nice" map is a weak equivalence/fibration iff it is a weak equivalence/fibration wrt Strom, equiv, Quillen, model structure. But for example, I do not know if a fibre bundle over a "nice" space with a "nice" fibre does necessarily have this stronger lifting property. $\endgroup$
    – user420620
    Jan 20, 2022 at 10:09

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