Center is closed in T0 topological group

Statement

The center of a T0 topological group is always a closed subgroup.

Facts used

1. Center is marginal
2. Marginal subgroup is closed in T0 topological group
3. Centralizer is closed in T0 topological group

Proof

Proof using marginal subgroup route

The proof follows directly from Facts (1) and (2).

Proof using centralizer route

• The centralizer of any element of the group is a closed subgroup (Fact (3))
• The center of the group is the intersection of all centralizers
• An arbitrary intersection of closed subsets of a topological space is closed