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.

  • $\begingroup$ +1 Interesting question. Out of curiosity: why do you ask for compact sets? Do you know an example for non-compact ones? $\endgroup$ – M. Winter Dec 6 '17 at 9:40
  • 2
    $\begingroup$ 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... $\endgroup$ – Ali Caglayan Dec 13 '17 at 0:00

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.

enter image description here

The above answer is written by @WlodekKuperberg

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.