Nonhomeomorphic subsets of the plane

general-topology compactness examples-counterexamples

This CW answer is supposed to kick this question from the unanswered queue. I strictly follow the approach mentioned in What to do with questions that are exact duplicates from MathOverflow?

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.

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The above answer is written by @WlodekKuperberg

MO link: Is it true that $X\times I\sim Y\times I\implies X\sim Y$?

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