I noticed that this question has been asked before, but I haven't been able to find a good answer for it, so here goes. For a course project I am learning about fibre bundles and fibrations, and now I am looking into some examples of both. Now, I would like to find an example of a fibration (either Hurewicz or Serre) that is not a fibre bundle. So far, I have only found the following (incomplete) example:
Let $E := [0, 1]^2 /l$, where $l$ is the line $\{\frac{1}{2}\} \times [0,1]$, and let $B = [0,1]$. Then a fibration $\pi: E \to B$ is not a fibre bundle.
However, I do not know how to define such a fibration $\pi$ explicitly, nor do I know how I would prove that it is a fibration (and whether it is Hurewicz or Serre). Moreover, I would not know how to prove that such a fibration is not a fibre bundle.
Therefore, I was wondering if anyone would have any hints regarding the aforementioned problems I have with this example. I would also like to know whether anyone knows any other examples of fibrations that are not fibre bundles. Thanks in advance for your help!
Let me give you another example with an explicit map:
Consider the subspace $E = \lbrace (x,y) \in \mathbb{R}^2 \mid \vert y \vert \leq \vert x \vert \rbrace \subset \mathbb{R}^2$. I claim that the projection onto the $x$-axis $$\text{pr} \colon E \rightarrow \mathbb{R}, (x,y) \mapsto x$$ is not a fiber bundle, but a (Hurewicz-)fibration. First of all, let me draw you a picture of the situation:
$\bullet$ Why is this not a fiber bundle?
Fiber bundles have homeomorphic fibers. Do you see where this fails?
$\bullet$ Why is this a fibration?
Let me give you a hint and leave the rest to you. I claim that the inclusion $E \hookrightarrow \mathbb{R}^2$ is a retraction and that this retraction allows you to steal the homotopy lifting property from the trivial fiber bundle $\mathbb{R}^2 \rightarrow \mathbb{R}$ (also the projection onto the first component).