Proposition 2

Let G be a finite-dimensional Lie group and H an integral subgroup of G. The following conditions are equivalent: (i) H is closed; (ii) the topology on H is induced by that on G; (iii) H is a Lie subgroup of G. (i) $\Rightarrow$ (iii): this follows from § 1, Propositions 2 (iv) (no. 1) and 14 (iii) (no. 7). (iii) $\Rightarrow$ (ii): obvious. (ii) $\Rightarrow$ (i): if the topology on H is induced by that on G, H is closed because H is complete (§ 1, no. 1, Proposition 1).