# Fibration with contractible fibre

Consider $f \colon E \rightarrow B$ a Serre fibration with contractible fibre (assume $B$ is path connected, so all fibres are weakly homotopy equivalent). Can we conclude from this that $f$ must be an homotopy equivalence?

I know that if we take additional assumptions on the involved spaces (like they have the homotopy type of CW complexes) the claim is true but I would like to understand if this holds in general or there is a counterexample.

Thanks in advance for any help.

Let $$W$$ be the Warsaw circle. This is the space that results by connecting the two loose ends of the topologists sine curce by a disjoint simple arc in $$\mathbb{R}^2$$. Note that $$W$$ is connected and path-connected, but not locally path-connected.
Now there is a regular Hurewicz fibration $$p:[0,1)\rightarrow W$$ whose fibres are single points. It is the obvious bijection. However $$p$$ is not a homotopy equivalence, since $$W$$ is not contractible (it admits essential maps into $$S^1$$).