\begin{definition}[Definition:P-adic Topology]
Let $\Z$ denote the set of integers.
Let $p \in \Z$ be a fixed prime number.
Let $\BB$ be the set of all sets $\map {U_\alpha} n$ defined as:
:$\map {U_\alpha} n = \set {n + \lambda p^\alpha: \lambda \in \Z}$
Then $\BB$ is the basis for a topology $\tau$ on $S$.
$\tau$ is referred to as the '''$p$-adic topology (on $\Z$)'''.
The topological space $T = \struct {S, \tau}$ is referred to as the '''$p$-adic (topological) space'''.
\end{definition}