Serre fibration vs Hurewicz fibration

Question

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.

Answer 1

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.

(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). 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.

Answer 2

These show up all the time in various generalized covering space theories, the reason being that you want homotopy lifting with respect to a certain class of spaces that you're interested in (e.g. locally path-connected spaces) which includes all cubes $[0,1]^n$ but probably not all spaces.

An important example with a simple construction is the generalized universal covering $p:\widetilde{\mathbb{H}}\to\mathbb{H}$ of the Hawaiian earring $\mathbb{H}$ that I describe in this answer. It is characterized by it's lifting property (described in link) which shows it to be a Serre fibration. It is not a Hurewicz fibration since it doesn't have homotopy lifting with respect to non-locally path-connected (even contractible) spaces like $\displaystyle\frac{[0,1]\times\{1,1/2,1/3,...,0\}}{\{0\}\times\{1,1/2,1/3,...,0\}}$.