请输入您要查询的字词:

 

单词 CenterOfAHausdorffTopologicalGroupIsClosed
释义

center of a Hausdorff topological group is closed


Theorem - Let G be a Hausdorff topological groupMathworldPlanetmath. Then the center of G is a closed normal subgroupMathworldPlanetmath.

Proof: Let Z be the center of G. We know that Z is a normal subgroup of G. Let us see that it is closed.

Let sZ¯, the closure of Z. There exists a net {sλ} in Z converging to s. Then, for every gG, we have that

  • gsλgs

  • sλgsg

But since Z is the center of G we have that gsλ=sλg, and as G is Hausdorff one must have sg=gs. This implies that sZ, i.e. Z is closed.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 18:39:26