---
tag: 0D939CMME7
kind: definition
name: Connected space
msc: [54D05]
defines: 0V032VWTHD
uses: [0CNQXVBHCX, 01ZQQDCERP, 0G5AV9Q4RR]
sources:
  - url: https://en.wikipedia.org/wiki/Connected_space
    locator: Definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is connected when there are no [[open sets|01ZQQDCERP]] $U, V \subseteq X$ with $U \neq \varnothing$, $V \neq \varnothing$, $U \cap V = \varnothing$ and $U \cup V = X$.

## Remarks

Equivalently, the only subsets of $X$ that are both open and [[closed|0G5AV9Q4RR]] are $\varnothing$ and $X$.
Equivalently, each [[continuous map|0SY9ECQ19Q]] from $X$ to the discrete space $\{0, 1\}$ is constant.
The interval $[0, 1]$ is connected. The set $[0, 1] \cup [2, 3]$ is not connected.
Under this definition the empty space is connected. Some authors require a connected space to be nonempty.

## Explanations

### L1 human

A connected space cannot be split into two separate open parts.
Connectedness is the topological form of the idea that a space is in one piece.