# cone

Given a topological space^{} $X$, the cone on $X$ (sometimes denoted by $CX$) is the quotient space^{} $X\times [0,1]/X\times \left\{0\right\}.$ Note that there is a natural inclusion $X\hookrightarrow CX$ which sends $x$ to $(x,1).$

If $(X,{x}_{0})$ is a based topological space^{}, there is a similar reduced cone construction, given by $X\times [0,1]/(X\times \left\{0\right\})\cup (\left\{{x}_{0}\right\}\times [0,1]).$ With this definition, the natural inclusion $x\mapsto (x,1)$ becomes a based map, where we take $({x}_{0},0)$ to be the basepoint of the reduced cone.

Title | cone |
---|---|

Canonical name | Cone |

Date of creation | 2013-03-22 13:25:20 |

Last modified on | 2013-03-22 13:25:20 |

Owner | antonio (1116) |

Last modified by | antonio (1116) |

Numerical id | 7 |

Author | antonio (1116) |

Entry type | Definition |

Classification | msc 54B99 |

Related topic | Suspension^{} |

Related topic | Join3 |

Defines | reduced cone |