Closed Lie subgroups

Question

Suppose that $\phi : G \rightarrow H$ is a lie group homomorphism. How do I show that $\ker(\phi)$ is a closed subgroup of $G$? In general, when we refer to $H \leq G$ being closed, do we mean that $H$ is closed under the topology associated with $G$? Assuming that $\phi$ is a continuous map, we have that $\phi^{-1}(1) = \ker(\phi)$, since the inverse of a closed set is closed, we have that $\ker(\phi)$ is closed. Is this correct reasoning?