Hello,

An open set of with the Zariski topology means that it is the cartesian product of open sets in , still with the Zariski topology. (It's the product topology)

So . We can define similarly

Where

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 arefinite

Hence is finite andcannotbe equal to K, since K is infinite. So

But , by de Morgan's law.

So

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 ><)