I know that a variety $X$ that can be parametrized using rational functions is irreducible. My question is:
What can we get removing or replacing the hypothesis of $X$ being a variety by, for example, $X$ being a quasi-affine(projective) variety?
For example, using the stereographic projection, one can parametrize the $n$-dimensional sphere $S^n$ minus one point. That's a quasi-affine (irreducible) variety, since its the complement of a point in the variety $S^n$.
Is there a general result for these cases?