Prove that if is infinite then two not empty open sets of (Zariski topology) always intersect.
So . We can define similarly
For and to intersect, we must have :
So we can fix an i and study the intersection.
Since and are open in the Zariski topology (and nonempty), it means that and are finite
Hence is finite and cannot be equal to K, since K is infinite. So
But , by de Morgan's law.
We know that for any sets A and B,
So and hence
It follows that a topological space with the Zariski topology cannot be separate.
Is it clear enough ? (do tell me if there is any mistake ><)
algebraic set. this is obvious because if is infinite, then and is a prime ideal of because is an integral domain. Q.E.D.
According to my field theory book it is defined as follows. Let be a commutative ring (assumed to be unitary). Define to be the set of all prime ideals of . Let define . A subset of is closed if and only if it has the form . And this forms a topology on .