---
tag: 370PJ5BF70
kind: concept
name: Extremally disconnected space
msc: [54G05]
sources:
  - url: https://en.wikipedia.org/wiki/Extremally_disconnected_space
same_as:
  wikidata: Q1385462
  wikipedia: Extremally disconnected space
---

## Summary

An extremally disconnected space is a [[topological space|0CQHFPGJ45]] in which the [[closure|0QEAJB91GZ]] of every open set is open.
Equivalently, any two disjoint open sets have disjoint closures.
Compact Hausdorff extremally disconnected spaces are exactly the Stone spaces of complete [[Boolean algebras|0XADJKM699]].
By [[Gleason's theorem|CPZKZ2AV0G]], they are also the projective objects among compact Hausdorff spaces.

## Explanations

### L1 human

In most familiar spaces, the closure of an open set adds a boundary that is not open.
For example, the closure of $(0, 1)$ in $\R$ is $[0, 1]$, and $[0, 1]$ is not open.
In an extremally disconnected space this never happens: closures of open sets stay open.
Discrete spaces and the Stone–Čech compactification $\beta \N$ are examples.