\begin{definition}[Definition:Prime Ideal Topology]
Let $S$ be the set of all prime ideals $P$ of the integers $\Z$.
Let $\BB$ be the set of all sets $V_x$ defined as:
:$V_x = \set {P \in S: x \notin P}$
for all $x \in \Z_{>0}$.
Then $\BB$ is the basis for a topology $\tau$ on $S$.
$\tau$ is referred to as the '''prime ideal topology'''.
The topological space $T = \struct {S, \tau}$ is referred to as the '''prime ideal space'''.
\end{definition}