If at each rational number we take an open ball of constant size (that is the radius of the ball is the same for all rational numbers) then it seems that all these open balls cover $\mathbb{R}$ i.e, $\cup_{i=1}^\infty B(x_i,r) \supset \mathbb{R}$ (where $r > 0$ and $x_i$ are rationals).
If is the radius is not constant but arbitrary, then do the resulting open balls still cover $\mathbb{R}$ i.e., is it true that $\cup_{i=1}^\infty B(x_i,r_i) \supset \mathbb{R}$ (where $r_i > 0$)?
In general, is the above true for any separable metric space?
Thanks, Phanindra