There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.

The topology inherited from projective space coincides with the Zariski topology.

The map from the Chow ring to the cohomology ring is an isomorphism.

I'm looking for nice explanations of these two facts.

EDIT: by the topology inherited from projective space I do not mean the Zariski topology, but the classical topology (given by balls in the Euclidean metric on the affine space from which we construct projective space, say). Since I have no intuition for the Zariski topology in this embedding, I'm not even sure I've got the statement correct: one place where it occurs is on page 147 in Fulton's Young tableaux book.

EDIT 2: I have gotten it wrong; Fulton claims only that the Zariski closures of the open cells equal their classical closures. But he also calls it a general fact. I guess the proper version of my first question should be: **why are the Zariski closures of the open schubert cells equal to their classical closures in the Pluecker embedding, and is there a general fact from which this follows**?

whattopology on the projective space? – Mariano Suárez-Alvarez♦ Apr 3 '14 at 18:43