# loop space

Let $X$ be a topological space^{}, and give the space of continuous maps $[0,1]\to X$, the compact-open topology^{}, that is a subbasis for the topology is the collection of sets $\{\sigma :\sigma (K)\subset U\}$ for $K\subset [0,1]$ compact^{} and $U\subset X$ open.

Then for $x\in X$, let ${\mathrm{\Omega}}_{x}X$ be the subset of loops based at $x$ (that is $\sigma $ such that $\sigma (0)=\sigma (1)=x$), with the relative topology.

${\mathrm{\Omega}}_{x}X$ is called the loop space^{} of $X$ at $x$.

Title | loop space |
---|---|

Canonical name | LoopSpace |

Date of creation | 2013-03-22 12:15:26 |

Last modified on | 2013-03-22 12:15:26 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 54-00 |

Related topic | Suspension^{} |

Related topic | EilenbergMacLaneSpace |