# topological group (obsolete)

This entry is obsolete, having been superseded by a new entry (http://planetmath.org/TopologicalGroup2). It is being retained for a short while because of the attached thread.

A *topological group ^{}* is a triple $(G,\cdot ,\mathcal{T})$ where $(G,\cdot )$ is a group and $\mathcal{T}$ is a topology

^{}on $G$ such that under $\mathcal{T}$, the group operation

^{}$(x,y)\mapsto x\cdot y$ is continuous with respect to the product topology on $G\times G$ and the inverse map $x\mapsto {x}^{-1}$ is continuous on $G$.

Many authors require that the topology be Hausdorff^{}.

Title | topological group (obsolete) |
---|---|

Canonical name | TopologicalGroupobsolete |

Date of creation | 2013-03-22 12:12:54 |

Last modified on | 2013-03-22 12:12:54 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 22A05 |

Related topic | Group |

Related topic | TopologicalRing |

Related topic | BirkhoffKakutaniTheorem |

Related topic | CategoryOfPolishGroups |

Related topic | AlgebraicTopology |