Closed subgroup of a Lie group - Imported Knowlpedia Content

Let $G$ be a Lie group and let $H\le G$ be a subgroup.

Definition. $H$ is a closed subgroup if it is closed as a subset of the underlying manifold (equivalently, as a subset of the underlying Hausdorff topological space) of $G$ .

Remarks

Why this matters. Closedness is the exact hypothesis needed to ensure that $H$ inherits a canonical Lie group structure from $G$ : by the closed subgroup theorem, a closed subgroup is an embedded Lie subgroup. This is essential for forming smooth quotients such as the coset space $G/H$ , which becomes a manifold under the same hypothesis.