## You are here

Homeconnected space

## Primary tabs

# connected space

A topological space $X$ is said to be *connected* if there is no pair of nonempty subsets $U,V$ such that both $U$ and $V$ are open in $X$, $U\cap V=\emptyset$ and $U\cup V=X$. If $X$ is not connected, i.e. if there are sets $U$ and $V$ with the above properties, then we say that $X$ is *disconnected*.

Every topological space $X$ can be viewed as a collection of subspaces each of which are connected. These subspaces are called the *connected components* of $X$. Slightly more rigorously, we define an equivalence relation $\sim$ on points in $X$ by declaring that $x\sim y$ if there is a connected subset $Y$ of $X$ such that $x$ and $y$ both lie in $Y$. Then a connected component of $X$ is defined to be an equivalence class under this relation.

## Mathematics Subject Classification

54D05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections