Open balls in $p$-adic numbers. 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 $p^{i+1}$﻿

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.
