# uniform structure of a topological group

Let $G$ be a topological group^{}. There is a natural uniform structure on $G$ which induces its topology^{}. We define a subset $V$ of the Cartesian product $G\times G$ to be an entourage if and only if it contains a subset of the form

$${V}_{N}=\{(x,y)\in G\times G:x{y}^{-1}\in N\}$$ |

for some $N$ neighborhood^{} of the identity element^{}. This is called the *right uniformity* of the topological group, with which multiplication becomes a uniformly continuous map.
The *left uniformity* is defined in a fashion, but in general they don’t coincide, although they both induce the same topology on $G$.

Title | uniform structure of a topological group |
---|---|

Canonical name | UniformStructureOfATopologicalGroup |

Date of creation | 2013-03-22 12:47:21 |

Last modified on | 2013-03-22 12:47:21 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 10 |

Author | mps (409) |

Entry type | Derivation |

Classification | msc 54E15 |

Defines | right uniformity |

Defines | left uniformity |