Topology on p-adic integers from inverse limit and p-adic valuation

Question

I am aware of the question (Topology on $\mathbf{Z}_p$)

One can use two definitions of the p-adic integers, the one that sees them as infinite formal sums $\sum_{i=0}^\infty a_i p^i$ and the one that sees them as inverse limit $\varprojlim \mathbb Z/p^i\mathbb Z$. One can show that those two are isomorphic as rings.

Now the infinite formal sums can be endowed with the p-adic norm, which gives rise to a topological structure induced by the norm; and the inverse limit inherits the product topology (we give each group the discrete topology).

I am trying to show that those two topologies are equal.

So given a

The p-adic norm induces a metric in which a neighbourhood base of $0$ is given by the open balls $\{B(0,\frac{1}{p^n})\}$. Infinite formal sums in $B(0,\frac{1}{p^n})$ are divisible by $p^n$, and henceforth correspond to sequences in the inverse limit belonging to the subset $\left(\underbrace{\{0\}\times \{0\}}_{n\text{ times}}\times \prod_{k\geq n+1} \mathbb Z/p^k\mathbb Z\right)\cap \varprojlim \mathbb Z/p^i\mathbb Z$, obviously an open set.

Is this correct?

Conversely, an open basis set in $(\mathbb Z_p)_{\text{proj}}$ is of the form $$\left(\prod_{j\in K\subset \mathbb N\text{ finite}} U_j \times \prod_{j\in \mathbb N\setminus K} \mathbb Z/p^j\mathbb Z\right)\cap \varprojlim \mathbb Z/p^\ell \mathbb Z$$ where $U_j\subset \mathbb Z/p^j\mathbb Z$ is open .

How do I show that this corresponds to an open set in the p-adic metric?

Answer 1

Let $\Bbb Z_p$ denote the ring of $p$-adic ring as infinite formal sums. For every $n\in\Bbb N$ let $\varepsilon_n:\Bbb Z_p\to\Bbb Z/p^n\Bbb Z$ denote the reduction modulo $p^n$. Each of these is open and continuous, because for every $a\in\Bbb Z$ we have $$\varepsilon_n^{-1}\{a+p^n\Bbb Z\}=a+p^n\Bbb Z_p$$ Since the projective limit is endowed with the initial topology, the induced ring homomorphism $$\varepsilon:\Bbb Z_p\xrightarrow\sim\varprojlim\Bbb Z/p^n\Bbb Z$$ is continuous as well. Let $\lambda_n:\varprojlim\Bbb Z/p^n\Bbb Z\to\Bbb Z/p^n\Bbb Z$ denote the limit cone. Then \begin{align} \varepsilon\{a+p^n\Bbb Z_p\} &=\varepsilon\varepsilon_n^{-1}\{a+p^n\Bbb Z\}\\ &=\varepsilon^{-1}\varepsilon\lambda_n^{-1}\{a+p^n\Bbb Z\}\\ &=\lambda_n^{-1}\{a+p^n\Bbb Z\} \end{align} thus $\varepsilon$ is an open mapping hence an homeomorphism.