Prove that in the metric space $(\Bbb N ,d)$, where we define the metric as follows: let $m,n \in \Bbb N$ then, $$d(m,n) = \left|\frac{1}{m} – \frac{1}{n}\right|.$$ Then show that each singleton set is open.
I am facing difficulty in viewing what would be an open ball around a single point with a given radius?
Can I take the open ball around an natural number $n$ with radius $\frac{1}{2n(n+1)}$??
Best Answer
Consider $$K=\left\{ \frac 1 n \,\middle|\, n\in\mathbb N\right\}$$ equipped with the standard metric $d_K(x,y) = |x-y|$. Then $(K,d_K)$ is isometric to your space $(\mathbb N, d)$ via $\mathbb N\to K, n\mapsto \frac 1 n$. Open balls in $(K, d_K)$ are easy to visualize, since they are just the open balls of $\mathbb R$ intersected with $K$. This should give you an idea how the open balls in $(\mathbb N, d)$ look.