---
tag: DQ4E55T3BW
kind: concept
name: Clopen set
msc: [54A05, 54D05]
sources:
  - url: https://en.wikipedia.org/wiki/Clopen_set
same_as:
  wikidata: Q320369
  wikipedia: Clopen set
---

## Summary

A clopen set in a [[topological space|0CQHFPGJ45]] is a subset that is both [[open|02BH3MCBQB]] and [[closed|0ZCK8DQYN1]].
The empty set and the whole space are always clopen.
A space is [[connected|0V032VWTHD]] exactly when it has no other clopen sets, so clopen sets measure how a space splits into pieces.

## Explanations

### L1 human

Usually "open" and "closed" look like opposite properties, but a set can have both.
Picture a space made of two separate islands: each island is open, and its complement, the other island, is open too, so each island is also closed.
For example, in the space $[0, 1] \cup [2, 3]$ with the usual distance, the piece $[0, 1]$ is clopen.
In the real line $\R$ itself, only $\emptyset$ and $\R$ are clopen, because $\R$ is in one piece.