---
tag: 003QP7A8HG
kind: statement
role: proposition
name: Closure is the set of limits of sequences
msc: ["54E35"]
uses: [068P0P32DW, 06JYPJ4WHH, 09GQ3ZTQJ9]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_(topology)
    locator: Point of closure
---

## 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 there is a sequence $(s_n)$ of points of $S$ that [[converges|06JYPJ4WHH]] to $x$.

## Remarks

Proof of the direct part: by [[closure in a metric space|068P0P32DW]], for each $n \ge 1$ there is $s_n \in S$ with $d(x, s_n) < 1/n$. Then $s_n \to x$.
The converse follows because $\overline{S}$ is closed and [[closed sets contain their limits|09GQ3ZTQJ9]].
A consequence: $S$ is closed if and only if the limit of each convergent sequence of points of $S$ is in $S$.
In a general topological space, sequences are not enough, and one uses nets or filters instead.

## Explanations

### L1 human

In a metric space, the closure of a set consists of the limits of sequences that lie in the set.
For example, each real number is a limit of rational numbers, so the closure of $\mathbb{Q}$ is $\mathbb{R}$.