X a space. Zariski Topology on X vs. X with the usual topology

Question

Let X be a projective variety; In this case is the usual topology finer than the Zariski topology? Are all points in the Zariski topology closed in this case?

Let X= $\mathbb{P}^r_\mathbb{C}$. Do there exist non closed points in X if it is given the Zariski topology? If so, then they certainly can't be in X with the usual topology right?

What is the relationship between the two topologies for general X?

Answer 1

I'll answer on the assumption that you're talking about varieties rather than schemes...

All Zariski open/closed sets are open/closed in the "usual topology". But some open/closed sets in the "usual topology" are not open/closed in the Zariski topology. This is the opposite way round to what you said in the question.

For example, take $\mathbb A^2$, the affine 2-plane. The set $\{x = 0 \}$ is Zariski closed, being the vanishing set of the polynomial $x$. It is also closed under the usual topology. However, the set $\{ x^2 + y^2 \leq 1 \}$, which is closed under the usual topology, is NOT closed in the Zariski topology, since it is not the vanishing set of polynomials.

Notice that Zariski open sets are very big! They are so big that any two non-empty Zariski open sets have non-trivial overlap (assuming the variety is irreducible).

Every point is closed in the Zariski topology. A point with coordinates $(a_1, \dots, a_n)$ can be described as the vanishing locus of the polynomials $x - a_1, \dots , x - a_n$.

Perhaps what confused you is that you have heard people talking about "non-closed points" in the context of schemes. The difference between schemes and varieties is that the set of "points" on a scheme not quite what you expect - it is the set of irreducible closed subvarieties of the variety. Thus a scheme has more "points" than the corresponding variety! The "closed points" on a scheme are in correspondence with the genuine points on the varieties. The other, non-closed, "points" on the scheme are in correspondence with closed subvarieties of variety of dimension $\geq 1$, and the closure of a non-closed "point" on a scheme is the set of all closed subvarieties contained within the closed subvariety associated to the non-closed "point".