---
tag: 0B2305GNAZ
kind: statement
role: proposition
name: Open balls are open
msc: ["54E35"]
uses: [0DC3MPR7C9, 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 $r > 0$.
Then the [[open ball|0DC3MPR7C9]] $B(x, r)$ is an [[open set|01ZQQDCERP]] of $X$.

## Remarks

Proof: let $y \in B(x, r)$ and let $\varepsilon = r - d(x, y) > 0$. If $d(y, z) < \varepsilon$, then $d(x, z) \le d(x, y) + d(y, z) < r$ by the [[triangle inequality|01M6CW9VM6]]. Thus $B(y, \varepsilon) \subseteq B(x, r)$.
The open sets of $X$ are exactly the unions of open balls.

## Explanations

### L1 human

A point inside an open ball is at some distance from the edge of the ball.
A smaller ball about that point, with radius equal to this distance, stays inside the big ball.