$(\mathbb{Q},|\cdot|_{\mathbb{Q}})$ is not connected

Question

Hello i have this exercise:

let $(E,d)$ a connected metric space, with an unbounded metric, show that the sphere is not empty, after deduce that $(\mathbb{Q},|.|_{\mathbb{Q}})$ is not connected.

I prove the first question, but I have I problem with the second one, from the first question we deduce that if the sphere is empty then the space is not connected or the distance is bounded, but $|.|_{\mathbb{Q}} $ is the discrete distance so it is bounded, then how to deduce that $(\mathbb{Q},|.|_{\mathbb{Q}})$ is not connected?

Thank you

Answer 1

The symbol $|\cdot|_{\mathbb{Q}}$ is not a discrete metric. It is not a metric at all since it only takes 1 argument!

It is most likely a norm, probably the Euclidean one that induces a metric $d(x,y)=|x-y|_{\mathbb{Q}}$. Assuming that $|x|_{\mathbb{Q}}$ is the standard Euclidean norm, i.e. the absolute value then the induced metric is not bounded, since $d(n,0)=n$ for natural $n$.

And to conclude that $\mathbb{Q}$ is not connected it is enough to consider the sphere centered at $0$ with radius $\sqrt{2}$.