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 fibres to be discrete and we assume that H is connected and locally path-connected, does it still follow that the kernel is discrete?