Closed subgroup of a Lie group

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 $G$.

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.

Remark. The assumption cannot be dropped in general: non-closed subgroups can be dense and fail to be submanifolds, so there need not be a reasonable smooth structure making inclusion a smooth embedding.