# 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

then $H$ is closed.

Title locally closed subgroups of topological groups are closed LocallyClosedSubgroupsOfTopologicalGroupsAreClosed 2013-03-22 17:36:39 2013-03-22 17:36:39 asteroid (17536) asteroid (17536) 11 asteroid (17536) Theorem msc 22A05