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 ><)