The $p$-adics and Related Concepts

We may extend $| \cdot |_p$ to $\mathbb Q_p$ by setting $| [a_n] |_p = \lim |a_n |_p$. This limit necessarily converges, and provides an absolute value on $\mathbb Q_p$ (which we still denote by $|\cdot|_p$). This absolute value produces a metric $d_p(x, y) = | x – y |_p$ and from this we can construct the metric topology $\mathcal T$. The strong triangle inequality will then manifest itself in the fact that, in contrast to the metric topology on $\mathbb R$, $\mathcal T$ is totally disconnected. Here we cover the basic definitions and constructions of this topology and how it is similar and different to the usual topology on $\mathbb R$.

Recall that a *topology* on $\mathbb Q_p$ is a collection $\mathcal T$ of subsets of $\mathbb Q_p$ that we call* open*, such that $\mathcal T$ contains $\mathbb Q_p$ and the empty set and is closed under arbitrary unions and finite intersections. Topologies in topology are analogous to $\sigma$-algebras in measure theory (though of course their axioms differ).

Given a collection of subsets $\mathcal C$ of $\mathbb Q_p$ we set $\mathcal T(\mathcal C)$ to be the intersection of all topologies containing $\mathcal C$. The power set on $\mathbb Q_p$ is a topology, so this definition is non-vacuous. Usually $\mathcal C$ is described in terms of some basic neighborhoods around points in $\mathbb Q_p$. In this case we use the metric to define balls of fixed radius in $\mathbb Q_p$.

**Definition:** Given $x \in \mathbb Q_p$ and $r > 0$ we define the ball of radius $r$ about $x$ to be $$B(x; r) = \{ y \in \mathbb Q_p : |x – y|_p < r \}.$$ We will use these as basic open neighborhoods to generate the Borel topology $\mathcal T$.

Since we have open balls, we can also define closed balls, and we set $$\overline B(x; r) = \{ y \in \mathbb Q_p : |x – y|_p \leq r \}.$$ We have already run into one closed ball, $\mathbb Z_p$ is the closed unit ball, the closed ball of radius 1 about 0$.

It turns out the discrete nature of the $p$-adic absolute value means that open balls are in fact also closed and vice-versa. This condition is sometimes called *clopen*.

**Theorem: **For $r \in (p^n, p^{n+1}],$ $B(x, r) = \overline B(x, p^n)$. In particular $B(x, p^{n+1}) = \overline B(x, p^n)$ and hence all open neighborhoods are in fact clopen.

Besides being clopen, balls in $\mathbb Q_p$ are very different from balls in $\mathbb R$. We capture some of these differences in the following theorem.

**Theorem: **

- Every $y \in B(x, p^n)$ is at the center in the sense that $B(x, p^n) = B(y, p^n)$.
- If $B$ and $B’$ are two balls in $\mathbb Q_p$, then $B \cap B’$ is either empty or equal to one of $B$ or $B’$.
- The complement of a ball is clopen, and hence if $y$ and $x$ are in disjoint balls then there exists ball $B$ such that $x \in B, y \in B^c$ and $\mathbb Q_p = B \sqcup B^c$. That is $\mathbb Q_p$ is
*totally disconnected*(with respect to $\mathcal T$).

It is sometimes useful to introduce some more granular topologies on $\mathbb Q_p$. In particular, we call the topology $\mathcal T_n$ generated by all balls of radius $p^{-n}$ the $n$th *resolution* of $\mathbb Q_p$. The analog of these resolutions are uninteresting in $\mathbb R$ (why?), but here total disconnectedness implies that $\mathcal T_n$ are non-trivial new topologies which ignore details about $\mathbb Q_p$ that occur on scales below $p^{-n}$.

Finally, we observe that $\mathbb Q_p$ is locally compact and that $\mathbb Z_p$ is compact. This is important because locally compact abelian groups ($\mathbb Q_p$ and $\mathbb Z_p$ are groups under addition) are the subject of a beautiful theory that allows us to do analysis and probability on these spaces in a natural way.

**Theorem:** $\mathbb Q_p$ is *locally compact*. That is, given any $x \in \mathbb Q_p$ there exists an open neighborhood of $x$ contained in a compact set.