Note: this is just a watered down version of the Zariski topology on the spectrum of a commutative ring. I got the idea from Steen and Seecbach’s book Counterexamples in Topology.
I am presenting the idea while attempting to use as little ring theory as possible, as some of my students have not had abstract algebra as yet.
Consider the integers . An ideal is a subset of that consists of 0 and all multiples of a given integer. The smallest positive integer in an ideal is the generator of that ideal; we denote that ideal by .
Examples: since every integer is a multiple of 1, and .
An ideal is PRIME if it is generated by a prime number. Now if is a prime and because neither has as a prime factor. So the prime ideals are those ideals whose compliments are multiplicatively closed.
Consider , the set of prime ideals of . That is, The elements (points) of are prime ideals. Yes, is a prime ideal because if then or . is not a prime ideal because and a prime ideal cannot be all of .
Let’s create a basis for a topology on : let . We are indexing subsets of prime ideals by positive integers and zero. Now as every ideal contains zero. as no prime ideal contains . Now if , has a prime factorization which contains prime factors , so so . That is, the open basis elements are those collections of prime ideals that have a finite complement (those generated by the prime factorization of .
What is a closed set in this topology? A set is closed if there exists some ideal (not necessarily prime) such that . For example, is a closed set because is open. And note . Note: traditionally, the Zariski topology is defined in terms of closed sets.
Clearly a finite union of closed sets is closed and an arbitrary intersection of closed sets is closed.
Now this topology is irreducible, which means that every non-empty pair of open sets intersect as all of them contain . Remember: as every ideal contains 0. Hence, this topology is not Hausdorff nor is it ( a topology is if every two points lie in different open sets, though these sets may intersect each other). This does have the property of being in that, given two points, there is at least one open set that does not contain both of the points as for prime.