I'm working with the metric space $(\mathbb{N}, \rho)$ where $\mathbb{N}$ is the set of natural numbers and $\rho(x,y) = |\frac{1}{x} - \frac{1}{y}|$.
I'm considering the open balls on this metric. Are there any that are finite? Infinite? All of $\mathbb{N}$?
My hunch is that there are open balls that are finite and infinite. For example, the open ball $B(1, \frac{1}{2})$ seems to be just {$1$}.
But if we make the radius larger than $1$ doesn't the open ball becoming infinite?
Am I correct? Are there any open balls that are finite? Infinite? All of $\mathbb{N}$? Any other general statements we can make about the open balls?