For affine schemes, there are contravariant maps V(-) and I(-) that create a bijection between irreducible closed subsets of the affine scheme and prime ideals of the ring that is the ring of global sections of that scheme (i.e. points on the affine scheme).

What happens to V(-) and I(-) when we take an arbitrary scheme (not assuming it is Noetherian or quasicompact)?

In particular, given a point p of a scheme X, can we find an irreducible closed subset of X that corresponds to p, such that the correspondence is a bijection, the way we could with affine schemes?

sober spaces. The closed irreducible subset corresponding to a point is its closure. A more interesting bijection is the one between closed subschemes and quasicoherent sheaves of ideals. $\endgroup$ – Denis Nardin Mar 18 '18 at 15:09