How can I prove that Let we have the following ultrametric space $(z,|.|_2)$ such that if $x=r.2^n$ then $|x|_2=2^{-n}$ how can I prove that the topology produced by this metric isn't discrete topology ?
Let's try being more precise. You are defining an ultrametric $d$ on the set $\mathbb{Z}$ of integers by setting
$$
|x|_2=\begin{cases}
0 &amp; \text{if $x=0$}\\[3px]
2^{-n} &amp; \text{if $x\ne0$ and $x=2^nr$, with $r$ odd}
\end{cases}
$$
and $d(x,y)=|x-y|_2$.

Now consider the sequence $(2^n)_{n\in\mathbb{N}}$. Is it convergent?
Show that there is some $y\in\Bbb Z$ such that $\{y\}$ is not open. For a hint, hover over the spoiler box below:


  Note that $|x-0|_2=|x|_2$ for any $x\in\Bbb Z.$