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.

Title center of a Hausdorff topological group is closed
Canonical name CenterOfAHausdorffTopologicalGroupIsClosed
Date of creation 2013-03-22 18:01:48
Last modified on 2013-03-22 18:01:48
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 4
Author asteroid (17536)
Entry type Theorem
Classification msc 22A05