Open Subset of Euclidean Space

Let $n\in\mathbb{N}$ and let $U\subseteq \mathbb{R}^n$. We say that $U$ is open in $\mathbb{R}^n$ if for every point $x=(x_1,\dots,x_n)\in U$ there exists a real number $r>0$ such that every point $y=(y_1,\dots,y_n)\in\mathbb{R}^n$ satisfying

$$\sum_{i=1}^n (y_i-x_i)^2<r^2$$

also belongs to $U$.