Closed ball

Definition

Let $(X, d)$ be a metric space, let $x \in X$ and let $r \ge 0$ be a real number. The closed ball with centre $x$ and radius $r$ is the set $$\bar{B}(x, r) = \{ y \in X : d(x, y) \le r \}.$$

Remarks

The open ball $B(x, r)$ is a subset of $\bar{B}(x, r)$. In $\mathbb{R}$ with the usual metric, $\bar{B}(x, r)$ is the closed interval $[x - r, x + r]$. The closed ball is not always the closure of the open ball. In the discrete metric, $B(x, 1) = \{x\}$ is closed, but $\bar{B}(x, 1) = X$. For $r = 0$, the closed ball is $\{x\}$.

Explanations

L1 human

The closed ball contains the points at distance at most $r$, so it also contains its edge. In the plane, it is the disc of radius $r$ together with its circle.