Let $Y$ be any topological space. In my notes I found the exercise to show that: $I \times Y \approx S^n $ via a homeomorphism is not possible, where $S^n$ denotes the $n$-sphere and $I$ the unit interval.
It is used in the proof of the Jordan curve theorem, so maybe a proof without using this theorem would be appropriate.
Obviously $Y \simeq I \times Y \approx S^n$, hence $Y \simeq S^n $, but I guess this does not get me any further.
Thanks in advance.
If this were true, the sphere would also be homeomorphic to $[0,1/2] \times Y$, and hence you would have a subspace of $S^n$ homeomorphic to $S^n$ that isn't the whole space. This is impossible by invariance of domain. I'm sure there's a more elementary proof in the case $n=1$, but I expect $n>1$ probably wants a homological proof like this.
Of course, the same proof shows that $I \times Y$ cannot ever be homeomorphic to a closed manifold.