Marginal implies unconditionally closed

Statement
Suppose $$G$$ is a T0 topological group (i.e., a topological group whose underlying set is a T0 space) and $$H$$ is a marginal subgroup of $$G$$ as an abstract group. Then, $$H$$ is a closed subgroup of $$G$$ (i.e., it is a closed subset in the topological sense). In fact, $$H$$ is a closed normal subgroup of $$G$$.

In particular, the result applies to the cases that $$G$$ is a Lie group.

Applications

 * Center is closed in T0 topological group

Facts used

 * 1) uses::Marginal implies algebraic (the intermediate property used here is algebraic subgroup).
 * 2) uses::Algebraic implies unconditionally closed

Proof
The proof follows by combining Facts (1) and (2).