Suppose $\theta: G\to K$ is a homomorphism of topological groups. We know that $G$ and $K$ each have fundamental systems of open neighbourhoods of the identity.
If I want to show that $\theta$ is continuous then is it enough to show that $\theta^{-1}(H)$ is open in $G$ where $H$ is an open, normal subgroup of $K$? I am guessing this is true but I can't see how to prove it, using the definition of the fundamental system.
Many thanks in advance.