# restricted direct product

Let ${\{{G}_{v}\}}_{v\in V}$ be a collection of locally compact topological groups. For all but finitely many $v\in V$, let ${H}_{v}\subset {G}_{v}$ be a compact open subgroup of ${G}_{v}$. The restricted direct product^{} of the collection $\{{G}_{v}\}$ with respect to the collection $\{{H}_{v}\}$ is the subgroup^{}

$$G:=\left\{{({g}_{v})}_{v\in V}\in \prod _{v\in V}{G}_{v}\right|{g}_{v}\in {H}_{v}\text{for all but finitely many}v\in V\}$$ |

of the direct product^{} ${\prod}_{v\in V}{G}_{v}$.

We define a topology on $G$ as follows. For every finite subset $S\subset V$ that contains all the elements $v$ for which ${H}_{v}$ is undefined, form the topological group^{}

$${G}_{S}:=\prod _{v\in S}{G}_{v}\times \prod _{v\notin S}{H}_{v}$$ |

consisting of the direct product of the ${G}_{v}$’s, for $v\in S$, and the ${H}_{v}$’s, for $v\notin S$. The topological group ${G}_{S}$ is a subset of $G$ for each such $S$, and we take for a topology on $G$ the weakest topology such that the ${G}_{S}$ are open subsets of $G$, with the subspace topology on each ${G}_{S}$ equal to the topology that ${G}_{S}$ already has in its own right.

Title | restricted direct product |
---|---|

Canonical name | RestrictedDirectProduct |

Date of creation | 2013-03-22 12:35:38 |

Last modified on | 2013-03-22 12:35:38 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 5 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 11R56 |

Classification | msc 22D05 |