---
tag: 90PB1H41A6
kind: definition
name: Clopen set
msc: [54A05, 54D05]
defines: DQ4E55T3BW
uses: [0CQHFPGJ45, 02BH3MCBQB, 0ZCK8DQYN1]
sources:
  - url: https://en.wikipedia.org/wiki/Clopen_set
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]].
A subset $A \subseteq X$ is *clopen* if $A$ is [[open|02BH3MCBQB]] in $X$ and $A$ is [[closed|0ZCK8DQYN1]] in $X$.
Equivalently, $A$ and its complement $X \setminus A$ are both open.

## Remarks

Example: let $X = [0, 1] \cup [2, 3]$ with the topology from $\R$.
The set $A = [0, 1]$ equals $X \cap (-1, 3/2)$, so it is open in $X$.
Its complement $[2, 3]$ equals $X \cap (3/2, 4)$, so it is open as well, and $A$ is closed.

Example: in the rationals $\Q$ with the topology from $\R$, the set $\set{q \in \Q : q < \sqrt{2}}$ is clopen, since $\sqrt{2} \notin \Q$.

Non-example: in $\R$, the set $[0, 1)$ is not open, because no interval around $0$ lies inside it.
It is not closed either.

In every space, $\emptyset$ and $X$ are clopen.
A space $X$ is [[connected|0V032VWTHD]] if and only if these are its only clopen subsets.