Center is closed in T0 topological group

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

Facts used

 * 1) uses::Center is marginal
 * 2) uses::Marginal subgroup is closed in T0 topological group
 * 3) uses::Centralizer is closed in T0 topological group

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