locally closed subgroups of topological groups are closed

Let $G$ be a Hausdorff topological group and $H\subseteq G$ a subgroup (which is a topological group itself under the subspace topology).

Theorem - If $H$ is locally closed in $G$ then $H$ is closed.

In particular we see that if $H$ is either

• open, or

• discrete (http://planetmath.org/Discrete), or

• locally compact,

then $H$ is closed.