Nonhomeomorphic subsets of the plane

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.

• +1 Interesting question. Out of curiosity: why do you ask for compact sets? Do you know an example for non-compact ones? Dec 6 '17 at 9:40
• The closest example I have found is here: Let $X$ be the torus with a hole and $Y$ be a disc with two holes. Then $X \times [0,1] \approx Y \times[0,1]$ as they are both solids in $\Bbb R^3$ bounded by a sphere with two handles. However for obvious reasons $X$ is not a subset of the plane... Dec 13 '17 at 0:00

1 Answer

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.

The above answer is written by @WlodekKuperberg