center of a Hausdorff topological group is closed
Theorem - Let be a Hausdorff topological group. Then the center of is a closed normal subgroup
.
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
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But since is the center of we have that , and as is Hausdorff one must have . This implies that , i.e. is closed.