Continuous bijection from an open subset of $\mathbb{R}^2$ to $S^2$

Question

I want to find continuous bijective mapping from $U \subseteq \mathbb{R}^2$ (U open set) to $S^2 \subset \mathbb{R}^3$ if it exists.

I think it doesn't but can't find way to prove it. I can prove that there isn't a continuous mapping from $S^2$ to open subset of $\mathbb{R}^2$ but don't know how to use it.

I would appreciate any hint or answer.

Answer 1

By invariance of domain, any continuous injection between manifolds of the same dimension is an open map. In particular, a continuous bijection $U\to S^2$ would be open and thus its inverse would be continuous, which you say you know is impossible.