Question: For RxR to be finer than the Zariski topology, Every open set of the Zariski topology must be the union of one or more basic open sets of RxR (I hope). If this is true, then in RxR, the graph of every polynomial F is closed. If the graph of F is closed, then every sub-sequence of F (call it Fn) must converge to a point in Fn? (This is easy to argue, but needs to be proved). So following this silliness..

I need to show that every sub-sequence of every polynomial converges to a point of the respective polynomial (this is assuming continuous polynomials). Once I have this, and the proof that F is closed if and only if every sub-sequence Fn converge to some point of F. Then I can say, well give me a polynomial, and I'll give you a closed set in RxR. But if that set is closed, then the complement is open. Therefore I have an open set in RxR that is equal to an open set in Zariski. But this works for any polynomial, so for every open set in Zariski I can find an open set in RxR.. so Zariski is coarser than RxR BAM!

Soo.. before I start doing all this stuff, does it make sense? And if someone else tried that Zariski test question, how did you approach it?

And also.. a basic open set in Zariski is what, the complement of a singleton point? ie ~(0,0) or ~(6,-2) (where ~ is the complement).

Answer: A basic open set in the Zariski topology is the complement of the zero set of a polynomial.

As for an approach to the problem: Forget this sub-sequence horsepucky.

Show that every set which is open in the Zariski topology is open in the usual topology.

To do that, show that every basic open set in the Zariski topology is open in the usual topology.

Equivalently, show that every zero set of a polynomial is closed in the usual topology. (Note: the zero set of f is not the same as its graph.)

If f is a polynomial, then the zero set of f is f inverse of {0}.

I'll let you work with those hints . . .

Question Is the Zariski Topology connected in R2?

Answer On November 8 I put forth this question at "Ask A Topologist" and got a helpful reply the very next day. If you want to see the hint, look for it here