Let $G$ a connected topological group, consider $G \times G$ and denote with $\Delta$ the diagonal of $G \times G$. Fix $U$ a symmetric open neighborhood of $e_G$. Define $$D := \{(p,q) \in G \times G: pq^{-1} \in U\}$$
I want to prove that $D$ is connected. I first noticed that $D \supseteq \Delta $ and that $D$ is open since is the preimage of the map $(p,q) \to pq^{-1}$, hence $D$ is an open neighborhood of the diagonal. Then I rewrote $D$ as $$D = \bigcup\limits_{q \in G}\{q\}\times Uq$$
In this way $D$ can be seen as a union of disjoint connected set intersecting a connected set, i.e $\Delta$, but I'm unable to conclude from here.
Any help or hint would be appreciated.