What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological?


Sorry if the question is too elementary for MO.



Serre fibration vs Hurewicz fibration
A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration.


* M. Steinberger and J. West. *Covering homotopy properties of maps between CW complexes or ANRs*. Proc. Amer. Math. Soc. **92** (1984), 573–577, doi:[10.1090/S0002-9939-1984-0760948-6](https://doi.org/10.1090/S0002-9939-1984-0760948-6).


(The proof is corrected in R. Cauty. *Sur les overts des CW-complexes et les fibr'es de Serre*. Colloquy Math. **63** (1992), 1–7, [link](https://bibliotekanauki.pl/articles/967590)). No relationship between the covering map and the CW structures is required. This argues either that counterexamples are
pathological or that it is a special property for the total space of a Serre fibration with CW base space to be a CW complex, although it has the homotopy type of a CW complex if the fiber does.


