-adic topology
Let be a ring and an ideal in such that
Though not usually explicitly done, we can define a metric on by defining for a by where is the largest integer such that (well-defined by the intersection assumption
, and is taken to be the entire ring) and by , and then defining for any ,
The topology induced by this metric is called the -adic topology. Note that the number 2 was chosen rather arbitrarily. Any other real number greater than 1 will induce an equivalent
topology.
Except in the case of the similarly-defined -adic topology, it is rare that reference is made to the actual -adic metric. Instead, we usually refer to the -adic topology.
In particular, a sequence of elements in is Cauchy with respect to this topology if for any there exists an such that for all we have . (Note the parallel with the metric version of Cauchy, where plays the part analogous to an arbitrary ). The ring is complete with respect to the -adic topology if every such Cauchy sequence
converges
to an element of .