# semilocally simply connected

A topological space^{} $X$ is semilocally simply connected if, for every point $x\in X$, there exists a neighborhood $U$ of $x$ such that the map of fundamental groups^{}

$${\pi}_{1}(U,x)\u27f6{\pi}_{1}(X,x)$$ |

induced by the inclusion map^{} $U\hookrightarrow X$ is the trivial homomorphism^{}.

A topological space $X$ is connected, locally path connected, and semilocally simply connected if and only if it has a universal cover^{}.

Title | semilocally simply connected |

Canonical name | SemilocallySimplyConnected |

Date of creation | 2013-03-22 12:38:46 |

Last modified on | 2013-03-22 12:38:46 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 54D05 |

Classification | msc 57M10 |

Synonym | semilocally 1-connected |

Synonym | locally relatively simply connected |

Related topic | Connected2 |

Related topic | SimplyConnected |

Related topic | ConnectedSpace |

Related topic | LocallyConnected |