# 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 $X$, $U\cap V=\mathrm{\varnothing}$ 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

^{}.

Title | connected space |

Canonical name | ConnectedSpace |

Date of creation | 2013-03-22 12:00:11 |

Last modified on | 2013-03-22 12:00:11 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 16 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54D05 |

Related topic | SemilocallySimplyConnected |

Related topic | ExtremallyDisconnected |

Related topic | ExampleOfAConnectedSpaceWhichIsNotPathConnected |

Related topic | LocallyConnected |

Related topic | ProofOfGeneralizedIntermediateValueTheorem |

Related topic | AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2 |

Related topic | AConnectedNormalSpaceWithMoreThanOn |

Defines | connected |

Defines | connected components |

Defines | disconnected |

Defines | connectedness |