# Serre fibration vs Hurewicz fibration

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.

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 $\displaystyle\frac{[0,1]\times\{1,1/2,1/3,...,0\}}{\{0\}\times\{1,1/2,1/3,...,0\}}$.