Specialization (pre)order

In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces the order becomes trivial and is of little interest. The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in order theory. Consider any topological space X. The specialization preorder ≤ on X relates two points of X when one lies in the closure of the other. However, various authors disagree on which 'direction' the order should go. What is agreed is that if x is contained in cl{y}, (where cl{y} denotes the closure of the singleton set {y}, i.e. the intersection of all closed sets containing {y}), we say that x is a specialization of y and that y is a generalization of x; this is commonly written y ⤳ x. Unfortunately, the property "x is a specialization of y" is alternatively written as "x ≤ y" and as "y ≤ x" by various authors (see, respectively, and ). Both definitions have intuitive justifications: in the case of the former, we have x ≤ y if and only if cl{x} ⊆ cl{y}. However, in the case where our space X is the prime spectrum Spec R of a commutative ring R (which is the motivational situation in applications related to algebraic geometry), then under our second definition of the order, we have y ≤ x if and only if y ⊆ x as prime ideals of the ring R. For the sake of consistency, for the remainder of this article we will take the first definition, that "x is a specialization of y" be written as x ≤ y. We then see, x ≤ y if and only if x is contained in all closed sets that contain y. x ≤ y if and only if y is contained in all open sets that contain x.