Totally disconnected and normal in connected implies central

Statement
Suppose $$G$$ is a fact about::connected topological group and $$H$$ is a normal subgroup of $$G$$, such that $$H$$ is a totally disconnected space in the subspace topology (i.e., the connected components of $$H$$ are one-point subsets). Then, $$H$$ is a central subgroup of $$G$$, i.e., $$H$$ is contained in the fact about::center of $$G$$.

A case of particular interest is where $$H$$ is a discrete subgroup of $$G$$, i.e., a closed subgroup of $$G$$ that is a discrete space under the subspace topology. This particular case says that any discrete normal subgroup of a connected topological group is central. However, there are many examples of totally disconnected normal subgroups that are not discrete.

Related facts

 * Central implies normal
 * Normal not implies central
 * Normal of least prime order implies central
 * Cartan-Brauer-Hua theorem

Proof
Given: A connected topological group $$G$$, a normal subgroup $$H$$ that is totally disconnected in the subspace topology.

To prove: $$H$$ is contained in the center of $$G$$.

Proof: The idea is to look at a fixed but arbitrary element in $$H$$ and a continuously varying element of $$G$$ that conjugates on this element. Note that the construction used in this proof involves conjugation, but viewed in a different way from usual. In the usual way of thinking about conjugation, the conjugating element is kept fixed and the element being conjugated is varied. Here, the element being conjugated is kept fixed and the conjugating element is varied continuously.