Topology on $\mathbf{Z}_p$

I will give an elementary proof of something stronger : the equality of the two topologies.

Note (and it is elementary to prove) that $v_p : \mathbf{Z}_p \backslash\{0\} \to \mathbf{N}$ has the following properties : $v_p (xy) = v_p(x) v_p(y)$ and $v_p(x+y)\geq \inf(v_p(x), v_p(y))$ for each $x,y$ such that $v_p (xy)$ and $v_p(x+y)$ have a meaning. (If you want you can extend $v_p$ at $0$ by setting $v_p (0) = +\infty$.) This obviously implies that (the ring) $\mathbf{Z}_p$ is a domain.

Now, the ideals $p^n \mathbf{Z}_p$ form a fundamental system of neighbourhoods of $0$ in $\mathbf{Z}_p$ for the product topology. (By abuse of language I call product topology on $\mathbf{Z}_p$ the topology induced on $\mathbf{Z}_p$ by the product topology on the product $\prod_{n\in\mathbf{N}} \mathbf{Z}/p^n \mathbf{Z}$ of which $\mathbf{Z}_p$ is a subspace.) Indeed, let $V$ be a neighbourhood of $0$ for the product topology. $V$ contains therefore an open (for the product topology...) $U$ containing $0$. By definition, $U = \mathbf{Z}_p \cap U'$ where $U'$ is open in $\prod_{n\in\mathbf{N}} \mathbf{Z}/p^n \mathbf{Z}$ of which $\mathbf{Z}_p$ for the product topology. By definition, $U'$ is union of sets of the form $\prod_{n\in\mathbf{N}} A_n$ where for each $n$ the set $A_n$ is open in $\mathbf{Z}/p^n \mathbf{Z}$, that is, is just a subset of $\mathbf{Z}/p^n \mathbf{Z}$ as the latter is endowed with the discrete topology, and where $A_n \not= \mathbf{Z}/p^n \mathbf{Z}$ only for finitely many values of $n$. Obviously one can assume that $U'$ is of this form, $U' = \prod_{n\in\mathbf{N}} A_n$, with the hypothesis previously made on the $A_n$'s kept. Let $d$ be the last integer for which $A_d \not= \mathbf{Z}/p^d \mathbf{Z}$. Then $A_n = \mathbf{Z}/p^n \mathbf{Z}$ contains $p^d (\mathbf{Z}/p^n \mathbf{Z})$ when $n > d$, and then $U'$ contains $p^d \prod_{n\in\mathbf{N}} \mathbf{Z}/p^n \mathbf{Z}$, so that $U$ contains $p^d\mathbf{Z}_p$. We have shown that each neighbourhood of $0$ contains a suitable $p^d\mathbf{Z}_p$, showing therefore that the ideals $p^n \mathbf{Z}_p$ form indeed a fundamental system of neighbourhoods of $0$ in $\mathbf{Z}_p$. But $x\in p^d\mathbf{Z}_p$ if and only if $v_p(x)\geq n$, if and only if $d(0,x) \leq e^{-n}$ if and only if $d(0,x) < e^{-(n-1)}$ so that the $p^n \mathbf{Z}_p$ are open balls in the metric topology.

The operations (sum and product) are continuous in the various $\mathbf{Z}/p^n \mathbf{Z}$ as these rings have the discrete topology, so that the "compositions" of operations of $\mathbf{Z}_p$ with the projections $\mathbf{Z}_p \to \mathbf{Z}/p^n \mathbf{Z}$ are all continuous, and by definition of the product topology, this means that the operations of $\mathbf{Z}_p$ are continuous.

The continuity of the operations implies that translations in $\mathbf{Z}_p$ are continuous, and this implies that for each $x\in\mathbf{Z}_p$ the sets $x+p^n \mathbf{Z}_p$ (which is the open ball of center $x$ and radius $e^{-(n+1)}$ as well as the closed ball of center $x$ and radius $e^{-n}$) form a fundamental basis of neighbourhoods of $x$ in both topopolgies (product and metric) on $\mathbf{Z}_p$. This shows that the product topology and that the metric topology are in fact equal. (Not only equivalent.)

Remark. I have shown that $p^n \mathbf{Z}_p$ are neighbourhoods of $0$ in the product topology. They are in fact of course open in this topology. Indeed, we saw that they are open balls for the metric topology, and that the metric and product topology where equal. But can we prove this directly ? Yes ! $x \in p^n \mathbf{Z}_p$, then $x + p^n \mathbf{Z}_p \subseteq p^n \mathbf{Z}_p$, and $x + p^n \mathbf{Z}_p$ is a neighbourhood of $x$ in the product topology. (We saw this fact earlier, with continuity of translations.) This means that $x$ in in the interior of $p^n \mathbf{Z}_p$ for the product topology. As it is true for all $x$, this implies that $p^n \mathbf{Z}_p$ is open in the product topology. Actually, and it is simple algebra also, one can prove that $p^n \mathbf{Z}_p$ is the kernel of the $n$-coordinate $\mathbf{Z}_p \rightarrow \mathbf{Z} / p^n \mathbf{Z}$ which is obviously continuous, and as $\{0\}$ is open in the target, its inverse image, $p^n \mathbf{Z}_p$, is also open.