To prove $\Bbb Q_p$ is not connected

Question

I would like to prove $\Bbb Q_p$ is not connected.

My try : Firstly,$\Bbb Z_p$ is not connected because $\Bbb Z_p＝p\Bbb Z_p∪(\Bbb Z_p - p\Bbb Z_p)$ and both $p\Bbb Z_p$ and $(\Bbb Z_p - p\Bbb Z_p)$ are clopen. So, $\Bbb Z_p$ is not connected.

But I don't know any clopen sets of $\Bbb Q_p$. Thank you for your help.

Answer 1

Hint: In any ultrametric space (such as $\mathbb Q_p$), any closed ball is also an open set. Easy proof from the ultrametric inequality $d(x,z) \le \max\{d(x,y), d(y,z)\}$.

Not only is it not connected. It is "totally disconnected".