Let $X$ be any topological space. Is it possible to construct a certain subspace $E\subseteq X\times X$ of the Product space, such that the restriction of the trivial bundle map $\pi\colon X\times X\to X$ to $E$ is a fibration (Hurewicz fibration) but not a fibre bundle?
I thought that intuitively the subspace should be designed so that it makes the fibres "non-constant", but at the same time lifting of homotopies is still possible.
Since I am very new to the topic, it seems difficult for me to come up with such a subspace $E$.
My only idea so far was to consider $X\times X-\Delta_X$, because it seems to me that by removing the diagonal, the fibres would be "cut into two pieces" whence they wouldn't be constant anymore..., but I am still struggling to prove that this works/or that it doesn't work.
Does such a space $E\subseteq X\times X$? exist at all? How is the space $E$ constructed in case that it exists?
Thank You for Your help.