---
tag: 0R37DFJKFW
kind: definition
name: Connected set
msc: [54D05]
defines: 0SPE8WNWFV
uses: [0D939CMME7, 0D5YHQGDJY]
sources:
  - url: https://en.wikipedia.org/wiki/Connected_space
    locator: Definition
---

## Definition

Let $X$ be a [[topological space|0CNQXVBHCX]] and let $S \subseteq X$.
The set $S$ is connected when $S$, with the [[subspace topology|0D5YHQGDJY]], is a [[connected space|0D939CMME7]].

## Remarks

Equivalently, there are no open sets $U, V$ of $X$ with $S \subseteq U \cup V$, $S \cap U \cap V = \varnothing$, $S \cap U \neq \varnothing$ and $S \cap V \neq \varnothing$.
The [[connected subsets of the real line are the intervals|0B9ZQFMQNV]].
The closure of a connected set is connected. A union of connected sets with a common point is connected.

## Explanations

### L1 human

A subset is connected when it is in one piece, as a space of its own.
For example, the set $\mathbb{Q}$ of rationals is not connected in $\mathbb{R}$: the sets $(-\infty, \sqrt{2})$ and $(\sqrt{2}, \infty)$ split it.