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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.

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timon92
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  • Possibly related (or could be interesting): http://math.stackexchange.com/q/1057907, http://math.stackexchange.com/questions/1716804, http://math.stackexchange.com/questions/396608 – Watson Jan 13 '17 at 13:47
  • Related. –  Jan 13 '17 at 13:48
  • It doesn't look so hard I think the problem is very hard. There are counterexamples when you use $\mathbb{R}^n$ or $S^n$ instead of $I$. But the $I$ variant seems to be quite interesting. – freakish Jan 13 '17 at 13:51
  • Thanks for the comments. I skimmed through the threads you've posted and I would expect that there should exist some counterexample to my question. – timon92 Jan 13 '17 at 13:57
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    Looks hard. I suggest you cross-post this on MO. – DanielWainfleet Jan 13 '17 at 23:39
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    @Watson, yeah, I was going to do so but I forgot about it. I will do it now. – timon92 Jan 30 '17 at 21:24
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    http://mathoverflow.net/questions/260953/is-it-true-that-x-times-i-sim-y-times-i-implies-x-sim-y – timon92 Jan 30 '17 at 21:41
  • This was the first problem of the Saint Petersburg Topology Olympiad! http://mathcenter.spb.ru/nikaan/olympiad/problemseng.pdf – Andres Mejia Nov 30 '17 at 19:52

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