---
tag: 0YS40B38JD
kind: statement
role: proposition
name: Closed balls are closed
msc: ["54E35"]
uses: [05DSV40EB9, 0G5AV9Q4RR, 0A26X79RTB, 01M6CW9VM6]
sources:
  - url: https://en.wikipedia.org/wiki/Ball_(mathematics)
    locator: In general metric spaces
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $x \in X$ and let $r \ge 0$.
Then the [[closed ball|05DSV40EB9]] $\bar{B}(x, r)$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: let $y \notin \bar{B}(x, r)$ and let $\varepsilon = d(x, y) - r > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \ge d(x, y) - d(y, z) > r$ by the [[triangle inequality|01M6CW9VM6]]. Thus the complement of $\bar{B}(x, r)$ is open.
For $r = 0$, this shows that each one-point set $\{x\}$ is closed in a metric space.

## Explanations

### L1 human

A point outside the closed ball is at some positive distance beyond its edge.
A small ball about that point stays outside, so the outside is open.