---
tag: 068P0P32DW
kind: statement
role: proposition
name: Closure in a metric space
msc: ["54E35"]
uses: [0DRKT1QJR7, 0DC3MPR7C9, 0A26X79RTB]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_(topology)
    locator: Definitions
---

## Statement

Let $(X, d)$ be a [[metric space|063W5JMY85]], let $S \subseteq X$ and let $x \in X$.
Then $x$ is in the [[closure|0DRKT1QJR7]] $\overline{S}$ if and only if for each real $\varepsilon > 0$ there is $s \in S$ with $d(x, s) < \varepsilon$.

## Remarks

Equivalently, $x \in \overline{S}$ if and only if each [[open ball|0DC3MPR7C9]] $B(x, \varepsilon)$ meets $S$.
For nonempty $S$, this means $d(x, S) = 0$, where $d(x, S)$ is the [[distance from the point to the set|07TXYYVYZR]].
In $\mathbb{R}$, the point $0$ is in the closure of $\{1/n : n \ge 1\}$, because $1/n < \varepsilon$ for large $n$.

## Explanations

### L1 human

A point is in the closure of $S$ when there are points of $S$ as close to it as you want.
The point does not need to be in $S$ itself.