Standard topology doesn't match with the topology induced by the p-adic metric I want to show that the standard topology on $\Bbb Q$ doesn't match with the topology that is induced by the p-adic metric. For this I wanna show that for the open set $M:= \{x \in \Bbb Q: |x|_p <\frac{1}{p}\}$ there exist no ball set in the standard topology with only these elements. I know that in $M$ there are only elements that are divisible by $p^2$  in the counter. But now I am stucked....

In $\Bbb Q$ with the $p$-adic metric, $\lim_{n\to\infty}p^n=0$, but not in $\Bbb Q$ with the usual metric.
