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.

Title	locally closed subgroups of topological groups are closed
Canonical name	LocallyClosedSubgroupsOfTopologicalGroupsAreClosed
Date of creation	2013-03-22 17:36:39
Last modified on	2013-03-22 17:36:39
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	11
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 22A05