---
tag: 01CYW8S04D
kind: definition
name: Totally disconnected space
msc: [54D05]
defines: 0RJSBR9GW1
uses: [0R37DFJKFW]
sources:
  - url: https://en.wikipedia.org/wiki/Totally_disconnected_space
    locator: Definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is totally disconnected when each [[connected subset|0R37DFJKFW]] of $X$ has at most one point.

## Remarks

Equivalently, each connected component of $X$ is a single point.
Each [[discrete space|0S2TEHX4E1]] is totally disconnected. So are $\mathbb{Q}$ and $\mathbb{R} \setminus \mathbb{Q}$, as subspaces of $\mathbb{R}$.
The [[Cantor set|0BXNDZFQME]] is totally disconnected, although it is compact and has no isolated points.

## Explanations

### L1 human

A totally disconnected space falls apart into single points: no subset with two or more points is in one piece.
This does not mean that the space is discrete. The rationals are a good example.