Definition:Open Ball/Real Analysis

Definition
Let $n \ge 1$ be a natural number.

Let $\R^n$ denote a real Euclidean space

Let $\left\Vert{\cdot}\right\Vert$ denote the Euclidean norm.

Let $a \in \R^n$.

Let $R > 0$ be a strictly positive real number.

The open ball of center $a$ and radius $R$ is the subset:
 * $B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert < R}\right\}$

Also see

 * Definition:Closed Ball (Real Analysis)
 * Definition:Open Set (Real Analysis)