p-adic topology I'm studying about p-adic numbers and p-adic analysis. I want to know about definition of topology on $\mathbb{Z}_p$ and topology on $\mathbb{Q}_p$ whit respect to p-adic norm. I saw this topology in the book of "A course in arithmetic" by J.-P. Serre and in the book of "A course in p-adic analysis" by A. M. Robert, but both of them have used "projective system". And also in the book of "P-adic analysis compared whit Real" by Svetlana Katok, at first the writer has introduced $\mathbb{Z}_p$ and $\mathbb{Q}_p$, and then has introduced open ball and so on. **I want to know is there a good way that at first, we introduce a topology in $\mathbb{Z}$ whit respect to p-adic norm, then we introduce the induced topology for $\frac{\mathbb{Z}}{p\mathbb{Z}}$, after all of this we introduce the topology on $\mathbb{Z}_p$?** (And also by this way intorduced the p-adic topology of $\mathbb{Q}_p$?) Thanks a lot!

You can start with $\mathbb{Z}$ equipped with the $p$-adic norm. Then consider its completion as a metric space, which is $\mathbb{Z}_p$. Then you can consider $\mathbb{Z}_p$ as a ring extending $\mathbb{Z}$. Finally, the field of fractions of $\mathbb{Z}_p$ will be $\mathbb{Q}_p$ and the topologies in these algebraic structures are always induced by an extension of the $p$-adic norm on $\mathbb{Z}$. Since $\mathbb{Z}_p$ with the $p$-adic norm is already complete (indeed, compact), you cannot build $\mathbb{Q}_p$ from it by using only topology.