Homeomorphic open sets have homeomorphic open balls

Question

If $U$ and $V$ are open sets in $R^n$, how to prove that there are two open balls $B_U$ and $B_V$ in $U$ and $V$ such that these two open balls are homeomorphic?

Answer 1

Any two open balls in $\mathbb{R}^n$ are homeomorphic (to the whole space $\mathbb{R}^n$ and thus to each other too). And any non-empty open set contains an open ball.

$B(0,1)$ is homeomorphic to $\mathbb{R}^n$ using the homeomorphism $f(x) = \frac{x}{1+\|x\|}: \mathbb{R}^n \to B(0,1)$. The rest is scaling and translating.