According to Wikipedia:
Let $G$ be a covering group of $H$. The kernel $K$ of the covering homomorphism is just the fiber over the identity in $H$ and is a discrete normal subgroup of $G$.
It is easy to show that the kernel is a normal subgroup, but why is it discrete?
I know this would be true if the identity of $H$ was open, but I cannot show this (and I don't even know if it is true/the right way to prove that $K$ is discrete).
EDIT: If we assume that the definition of "cover space" does not require the fibers to be discrete and we assume that $H$ is connected and locally path-connected, does it still follow that the kernel is discrete?