## Franklin's Notes

The p-adic absolute value on $\mathbb{Q}$ $|\cdot|_p$ is induced by the p-adic valuation , and is defined as where $|0|_p$ is taken to be $0$ by definition. This endows the rational numbers with a metric defined by turning it into a metric space, so that the p-adic numbers can be defined as the completion of $\mathbb{Q}$ with respect to $|\cdot|_p$.

It follows form the properties of the p-adic valuation that this metric satisfies the ultrametric inequality This has a couple of strange implications on the topology of the p-adic numbers $\mathbb{Q}_p$. In particular:

• All triangles is $\mathbb{Q}_p$ are isoscles.

• All open balls in $\mathbb{Q}_p$ are disconnected.

• The only nonempty connected subsets of $\mathbb{Q}_p$ are singletons. (That is, $\mathbb{Q}_p$ is totally disconnected.)

• For any two open balls, either one contains the other or they are disjoint.

• Every closed ball is also open.

• A sequence $(a_n)$ is Cauchy iff $|a_{n+1}-a_n|\to 0$ as $n\to\infty$.

• A series converges if and only if its terms tend to zero.

• $\mathbb Z_p$ is compact.