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?