Let $U\subset\mathbb{R^n}$ be a open set and $\mathbb{S^n}$ the unit sphere of $\mathbb{R^{n+1}}$(i.e. $\mathbb{S^n}=\{x\in\mathbb{R^{n+1}}:||x||=1\}$).
How can I show that there's no homeomorphism between $U$ and $\mathbb{S^n}$?
My progress:
As image of connected set over a continuous function is connected and $\mathbb{S^n}$ is connected, if there was such homeomorphism, then $U$ would be connected as well.
Now, using the fact that all connected open sets of $\mathbb{R^n}$ is homeomorphic to $\mathbb{R^n}$, it suffice to show that there's no homeomorphism between $\mathbb{R^n}$ and $\mathbb{S^n}$.
I know that for all $p\in \mathbb{S^n}$, $\mathbb{S^n}- \{p\}$ is homeomorphic to $\mathbb{R^n}$.
Any help would be much appreciated!
Best Answer
This is a collection of remarks that should answer your question:
The most common tool to prove that open sets of different dimension cannot be homeomorphic is invariance of domain. See the section on consequences.
Two open sets on manifolds of different dimensions cannot be homeomorphic either. The reason is that if you study the restriction of the homeomorphism to a small set, you get a homeomorphism between Euclidean open sets via coordinate charts. (This remark will only make sense if you are familiar with topological or smooth manifolds.)
However, now your two sets have the same dimension, so this method doesn't work.