If $u_1$, $u_2 \in U$ then product $u_1^{-1}u_2 \in U$?

Question

Suppose $G$ is a connected topological group and $H$ is a discrete normal subgroup of $G$, then $H$ is contained in the center of $G$.

Since $H$ is discrete, the identity element is not a limit point of $H$ and so there is a neighborhood $U$ of the identity such that $U \cap H = \{1\}$. I want to utilize the fact that $U$ has the property that if $u_1$, $u_2$ are in $U$ then the product $u_1^{-1}u_2$ is in $U$. My question is, how do I see this fact is true?

Answer 1

This fact is not true. Think of $G=\mathbf R $, $H=\mathbf Z $ and $U=(-1, 1) $.