Open balls in $p$-adic numbers.

Question

I am new to $p$-adic numbers and was watching an introductory video about it. At 14:51, he says that $r$ only takes values in the form of $p^n$. However, I don't understand why $r$ must be restricted to numbers in the form of $p^n$. Given any r that is not in the form of $p^n$, there will be an integer $i$ such that $p^i<r<p^{i+1}$. Thus, the open ball set of $r$ would be the same as $p^{i+1}$

Answer 1

As in any metric space, you can define an open ball of radius any real number. But if the metric only takes values that are powers of $p$, we might as well restrict attention to balls of radius a power of $p$. I suspect that this was precisely the point the video was trying to make.

Answer 2

Here is a simpler example. In a discrete metric space $X$ where each $d(x,x)=0$ and $d(x,y)=1$ for $x \ne y$ it makes sense to talk about balls of any radius. However there are really only two types of balls at each point. Namely $B(x,r)=\{x\}$ for $r=0$ and $B(x,r)=X$ otherwise. So we don't lose anything if we decide to only consider balls of radius $0$ and $5$ say. Likewise for the adic numbers, every $B(x,r)$ is equal to some $B(x,p^n)$ so we don't lose anything by only talking about the latter set of balls.