center of a Hausdorff topological group is closed
Proof: Let be the center of . We know that is a normal subgroup of . Let us see that it is closed.
Let , the closure of . There exists a net in converging to . Then, for every , we have that
But since is the center of we have that , and as is Hausdorff one must have . This implies that , i.e. is closed.
|Title||center of a Hausdorff topological group is closed|
|Date of creation||2013-03-22 18:01:48|
|Last modified on||2013-03-22 18:01:48|
|Last modified by||asteroid (17536)|