I-adic topology

Remarks

The cosets $x + I^n M$, for $x \in M$ and $n \ge 0$, form a base of this topology; each of them is open and also closed. Addition on $M$ and multiplication $R \times M \to M$ are continuous, so $R$ becomes a topological ring and $M$ a topological module. The topology is Hausdorff if and only if $\bigcap_n I^n M = 0$; then $d(x, y) = 2^{-\nu(x - y)}$, with $\nu(z)$ the largest $n$ such that $z \in I^n M$ and $\nu(0) = \infty$, is a metric for it. Example: on $\Z$ with $I = (2)$, the sequence $2^n$ tends to $0$, since $2^m \in 2^n \Z$ for all $m \ge n$. Non-example: the set $\set{0}$ is not open in $\Z$ for this topology, since no $2^n \Z$ is contained in $\set{0}$. So the $2$-adic topology on $\Z$ is not discrete, unlike the topology that $\Z$ gets from $\R$.