[Math] Are all the open sets in a Euclidean space homeomorphic

Question

I know that open balls are homeomorphic to the entire Euclidean space, and any convex open set can be proved to be homeomorphic to the entire Euclidean space. So I was wondering if all the open sets in $\mathbb{R}^n$ are homeomorphic?

Answer 1

No. For example, the open annulus $$\{(x, y) \in \Bbb R^2 : 1 < x^2 + y^2 < 2\}$$ is not simply connected, but the open ball $$\{(x, y) \in \Bbb R^2 : x^2 + y^2 < 1\}$$ is.