Discrete Normal Subgroup of Connected Group is Contained in Center

Theorem
Let $G$ be a connected topological group.

Let $\map Z G$ be the center of $G$.

Let $N$ be a discrete normal subgroup of $G$.

Then:
 * $N \subseteq \map Z G$

Proof
Let $h \in N$.

We shall show $h \in \map Z G$

Now, let:
 * $A _h := \set {g \in G : g^{-1} h g = h}$

Then, we need to show:
 * $A _h = G$

Let $e \in G$ denote the identity.

Since $e \in A _h$, we have $A _h \ne \O$.

Thus, it suffices to show that $A _h$ is clopen in view of.

As $N$ is a discrete subgroup, there is an open set $U \subseteq G$ such that:
 * $U \cap N = \set e$

Moreover, for each $h \in N$, $h U$ is an open set such that:
 * $h U \cap N = \set h$