---
tag: 0K67Q60Z8P
kind: statement
role: proposition
name: Complement of an open set is closed
msc: [54A05]
uses: [0G5AV9Q4RR, 01ZQQDCERP]
sources:
  - url: https://en.wikipedia.org/wiki/Closed_set
    locator: Definition
---

## Statement

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $U \subseteq X$.
Then $U$ is [[open|01ZQQDCERP]] in $X$ if and only if $X \setminus U$ is [[closed|0G5AV9Q4RR]] in $X$.

## Remarks

Proof: $X \setminus (X \setminus U) = U$, so the claim is the definition of a closed set applied to $X \setminus U$.
Thus the closed sets determine the open sets, and the open sets determine the closed sets.

## Explanations

### L1 human

Taking the complement turns open sets into closed sets and closed sets back into open sets.
For example, $(0, 1)$ is open in $\mathbb{R}$, so $(-\infty, 0] \cup [1, \infty)$ is closed.