Definition:Closed Ball

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $a \in A$.

Let $\epsilon \in \R: \epsilon > 0$ be a positive real number.

The closed $\epsilon$-ball of $a$ in $M$ is defined as:


 * $B^- \left({a; \epsilon}\right) := \left\{{x \in A: d \left({x, a}\right) \le \epsilon}\right\}$

If it is necessary to show the metric itself, then the notation $B_d^- \left({a; \epsilon}\right)$ can be used.

Radius
In $B^- \left({a; \epsilon}\right)$, the value $\epsilon$ is referred to as the radius of the ball.