Serre fibrations vs. Hurewicz fibrations

Question

What is an example of a Serre fibration that is not a Hurewicz fibration?

Answer 1

Let X be a mapping cylinder of the obvious map between the subspaces of the real line $\{n | n \in N\} \rightarrow {0}\cup\{1/n |n \in N^+\}$. Then the obvious map $X \rightarrow ({0}\cup\{1/n |n \in N^+\})\times I$ meets the requirement.

Denote $({0}\cup\{1/n |n \in N^+\})\times I$ by $Y$, and ${0}\cup\{1/n |n \in N^+\}$ by $Z$. So we have $f : X \rightarrow Y$.

Take the obvious map $k : Z \rightarrow X$. And an identity map $1_Y : Z\times I \rightarrow Y$ considered as a homotopy. Then the homotopy lifing fails.

On the other hand, it should not be hard to show that $f$ is a Serre fibration.