Definition:Prime Ideal Topology
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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 $S$.
$\tau$ 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
- Results about the prime ideal topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $56$. Prime Ideal Topology