---
tag: 05DSV40EB9
kind: definition
name: Closed ball
msc: ["54E35"]
defines: 0Q0H13G8ZB
uses: [063W5JMY85]
sources:
  - url: https://en.wikipedia.org/wiki/Ball_(mathematics)
    locator: In general metric spaces
---

## Definition

Let $(X, d)$ be a [[metric space|063W5JMY85]], 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|0DC3MPR7C9]] $B(x, r)$ 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|0NA4FK7B8S]], $B(x, 1) = \{x\}$ 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.