Is every nonempty open set in $\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$?

Question

Is every nonempty open set $U\subseteq\mathbb{R}^n$ diffeomorphic to $\mathbb{R}^n$?

I think this is false; perhaps I can take $U$ to be the disjoint union of two open balls. But how can I prove that this is not diffeomorphic to $\mathbb{R}^n$?

More generally, do diffeomorphisms preserve connectedness, or number of connected components?

Answer 1

For two sets to be diffeomorphic, they must be homeomorphic. Connectedness (and number of connected compotents) is a topological property and is therefore preserved under homeomorphism. Therefore $\mathbb{R}^n$ is not homeomorphic to any disconnected open subset of itself.

Answer 2

Diffeomorphism saves connectedness. But here $\mathbb R^n$ is connected and union of two disjoint open sets is not connected.