Let $B_d(x,r) \cap G_1$ where $G_1$ is an open dense set. Now this intersection is non empty. Let $x_1 \in B_d(x,r) \cap G_1$.Then, $B_d(x_1,r_1) \subset G_1$ .Now my doubt is that will there exists a $r_2 > 0$ such that $B_d(x_1,r_2) \subset G_1$ and $B_d(x_1,r_2) \subset B_d(x,r)$.
I think it will be true. However I was wondering if there is a contradicting example.