Immersed, embedded, and closed Lie subgroups

An immersed Lie subgroup of a Lie group $G$ is a Lie group $H$ together with an injective smooth group homomorphism $i:H\to G$ that is an immersion. When no adjective is printed, “Lie subgroup” means an immersed Lie subgroup. It may be identified with the set $i(H)$ only if that set is remembered with the intrinsic smooth-manifold topology transported from $H$; this topology need not equal the subspace topology from $G$.

The subgroup is embedded if $i$ is a smooth embedding, so its intrinsic topology is the subspace topology. It is closed if $i(H)$ is closed as a subset of $G$. These adjectives refer to the specified immersed subgroup; closedness alone does not silently replace its given intrinsic structure.

The definition permits $H=\{e\}$, $H=G$, disconnected subgroups, and zero-dimensional subgroups. Unless stated otherwise, all groups here are the finite-dimensional real Lie groups fixed by Lie group.