Marginal implies unconditionally closed

Statement

Suppose $G$ is a semitopological group 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 topological group, Lie group, or algebraic group.

Related facts

Applications

• Center is closed in semitopological group