# universal covering space

Let $X$ be a topological space^{}. A universal covering space is a covering space $\stackrel{~}{X}$ of $X$ which is connected and simply connected.

If $X$ is based, with basepoint $x$, then a based cover of $X$ is cover of $X$ which is also a based space with a basepoint ${x}^{\prime}$ such that the covering is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of $x$.

The universal covering space has the following universal property^{}: If $\pi :(\stackrel{~}{X},{x}_{0})\to (X,x)$ is a based universal cover, then for any connected based cover ${\pi}^{\prime}:({X}^{\prime},{x}^{\prime})\to (X,x)$, there is a unique covering map ${\pi}^{\prime \prime}:(\stackrel{~}{X},{x}_{0})\to ({X}^{\prime},{x}^{\prime})$ such that $\pi ={\pi}^{\prime}\circ {\pi}^{\prime \prime}$.

Clearly, if a universal covering exists, it is unique up to unique isomorphism^{}. But not every topological space has a universal cover. In fact $X$ has a universal cover if and only if it is semi-locally simply connected (for example, if it is a locally finite^{} CW-complex^{} or a manifold).

Title | universal covering space |
---|---|

Canonical name | UniversalCoveringSpace |

Date of creation | 2013-03-22 12:15:34 |

Last modified on | 2013-03-22 12:15:34 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 7 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 54-00 |

Synonym | universal cover |

Related topic | OmegaSpectrum |