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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!

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    Is that not just a case of defining the $p$-adic metric on $\mathbb{Q}$ and defining $\mathbb{Q}_p$ as its completion? – 2017-02-27
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    Yes. We can define $\mathbb{Q}_p$ as a completion of $\mathbb{Q}$ (by using cauchy sequence). But I want to introduce the topology at first on the $\mathbb{Z}_p$. – 2017-02-27
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    Im still slightly confused. Do you mean then why is the profinite topology of $\mathbb{Z}_p$ arising from being defined as the inverse limit of $\mathbb{Z}/p^k\mathbb{Z}$ equal to the $p$-adic one? – 2017-02-27
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    No, I want to know a different way of introducing topology of $\mathbb{Z}_p$ from these tow ways. – 2017-02-27

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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.

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    Is there a good reference, that I can learn this way from it? – 2017-02-28
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    Yes there is!: [Schikhof, Wilhelmus Hendricus.] Ultrametric Calculus: an introduction to p-adic analysis. Vol. 4. Cambridge University Press, 2007. But personally, I prefer the construction of $\mathbb{Q}_p$ as the completion of $\mathbb{Q}$ with the $p$-adic norm (valuation). – 2017-02-28