---
tag: 02TZDKG50Q
kind: statement
role: proposition
name: Closure is closed
msc: [54A05]
uses: [0DRKT1QJR7, 09YMV8H4X5]
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 the [[closure|0DRKT1QJR7]] $\overline{S}$ is a [[closed set|0G5AV9Q4RR]] of $X$.

## Remarks

Proof: $\overline{S}$ is an intersection of closed sets, and each [[intersection of closed sets is closed|09YMV8H4X5]].
It follows that $\overline{\overline{S}} = \overline{S}$.

## Explanations

### L1 human

The closure is built as an intersection of closed sets, and such an intersection is again closed.