I'm trying to find two compact, nonhomeomorphic subsets of the plane, say $X$ and $Y$, such that $X \times [0,1]$ is homeomorphic to $Y \times [0,1]$. I can not think of how a homeomorphism arises when you product with the interval.
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There are indeed counterexamples to which Igor Belegradek gave a reference. Here is another counterexample in the plane, perhaps the simplest there is: Let $X$ be an annulus with one arc attached to one of its boundary components and another arc attached to the other boundary component, and $Y$ - an annulus with two disjoint arcs attached to the same one of its boundary components.
The above answer is written by @WlodekKuperberg