Specialization preorder

Summary

The specialization preorder of a topological space orders its points by closure: $x \le y$ when $x$ lies in the closure of $\set{y}$. It is a partial order exactly when the space is T0, and it is the equality relation exactly when the space is T1. In algebraic geometry it records which points of a scheme specialize to which. In domain theory it recovers the order of a poset from its Scott topology, and in general it links topological spaces with ordered sets.