# quotient space

Let $X$ be a topological space^{}, and let $\sim $ be an equivalence relation^{} on $X$. Write ${X}^{*}$ for the set of equivalence classes^{} of $X$ under $\sim $. The quotient topology on ${X}^{*}$ is the topology^{} whose open sets are the subsets $U\subset {X}^{*}$ such that

$$\bigcup U\subset X$$ |

is an open subset of $X$. The space ${X}^{*}$ is called the quotient space of the space $X$ with respect to $\sim $. It is often written $X/\sim $.

The projection map $\pi :X\u27f6{X}^{*}$ which sends each element of $X$ to its equivalence class is always a continuous map^{}. In fact, the map $\pi $ satisfies the stronger property that a subset $U$ of ${X}^{*}$ is open if and only if the subset ${\pi}^{-1}(U)$ of $X$ is open. In general, any surjective map $p:X\u27f6Y$ that satisfies this stronger property is called a quotient map, and given such a quotient map, the space $Y$ is always homeomorphic to the quotient space of $X$ under the equivalence relation

$$x\sim {x}^{\prime}\iff p(x)=p({x}^{\prime}).$$ |

As a set, the construction of a quotient space collapses each of the equivalence classes of $\sim $ to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map $\pi $ is continuous^{}.

For $A\subset X$, one often writes $X/A$ for the quotient space obtained by identifying all the points of $A$ with each other.

Title | quotient space |
---|---|

Canonical name | QuotientSpace |

Date of creation | 2013-03-22 12:39:40 |

Last modified on | 2013-03-22 12:39:40 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54B15 |

Related topic | AdjunctionSpace |

Defines | quotient topology |

Defines | quotient map |