The following question came to my mind:
Let $I$ be the unit interval. Suppose that $X$ and $Y$ are topological spaces such that $X \times I$ is homeomorphic to $Y \times I$. Does it follow that $X$ is homeomorphic to $Y$?
It doesn't look so hard, but somehow I couldn't find a proof or counterexample.
If the answer turns out to be negative, I would be interested in what one should assume about $X$ and $Y$ to make the implication true.
Now reposted at MathOverflow.