---
tag: 0D2W7FS52T
kind: statement
role: corollary
name: A set is closed iff it equals its closure
msc: [54A05]
uses: [0SHC6Z304Z]
sources:
  - url: https://en.wikipedia.org/wiki/Closure_(topology)
    locator: Properties
---

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
Then $S$ is [[closed|0G5AV9Q4RR]] in $X$ if and only if $\overline{S} = S$.

## Remarks

Proof: if $S$ is closed, then $\overline{S} \subseteq S$ by [[closure is the smallest closed superset|0SHC6Z304Z]], and $S \subseteq \overline{S}$ always holds. If $\overline{S} = S$, then $S$ is closed because the [[closure is closed|02TZDKG50Q]].
The dual statement is: $S$ is open if and only if $\operatorname{int} S = S$.

## Explanations

### L1 human

A closed set already contains all the points that you can approach from inside it.
So taking the closure adds nothing.