 Schemes are Kolmogorov spaces

The underlying topological space of every scheme is a Kolmogorov space: any two distinct points are distinguished by an open subset containing just one. Choose an affine open subscheme containing the first point. If it omits the second, it already distinguishes them. Otherwise, the two points correspond to distinct prime ideals, and an element in one but not the other defines a principal open subscheme distinguishing them. Thus distinct points of a scheme have distinct closures.