Definition:Prime Ideal Topology

Definition
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 $S$.

Thus the sets of the form $\openint a b$ such that $a < 0 < b$ are open sets in $S$.

$\tau$ is referred to as the prime ideal topology.

The topological space $T = \struct {S, \tau}$ is referred to as the prime ideal space.

Also see

 * Prime Ideal Topology is Topology


 * Existence of Topological Space which satisfies no Separation Axioms but $T_0$