Marginal implies unconditionally closed

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., marginal subgroup) must also satisfy the second subgroup property (i.e., unconditionally closed subgroup)
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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.

Related facts

Applications

• Center is closed in T0 topological group

Facts used

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

Proof

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