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.
Definition and motivation
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.
These restatements help to explain why one speaks of a "specialization": y is more general than x, since it is contained in more open sets. This is particularly intuitive if one views closed sets as properties that a point x may or may not have. The more closed sets contain a point, the more properties the point has, and the more special it is. The usage is consistent with the classical logical notions of genus and species; and also with the traditional use of generic points in algebraic geometry, in which closed points are the most specific, while a generic point of a space is one contained in every nonempty open subset. Specialization as an idea is applied also in valuation theory.
The intuition of upper elements being more specific is typically found in domain theory, a branch of order theory that has ample applications in computer science.
Upper and lower sets
Let X be a topological space and let ≤ be the specialization preorder on X. Every open set is an upper set with respect to ≤ and every closed set is a lower set. The converses are not generally true. In fact, a topological space is an Alexandrov-discrete space if and only if every upper set is also open (or equivalently every lower set is also closed).
Let A be a subset of X. The smallest upper set containing A is denoted ↑A
and the smallest lower set containing A is denoted ↓A. In case A = { x} is a singleton one uses the notation ↑x and ↓x. For x ∈ X one has:
↑x = { y ∈ X : x ≤ y} = ∩{ open sets containing x}. ↓x = { y ∈ X : y ≤ x} = ∩{ closed sets containing x} = cl{ x}.
The lower set ↓x is always closed; however, the upper set ↑x need not be open or closed. The closed points of a topological space X are precisely the minimal elements of X with respect to ≤.
Examples
In the Sierpinski space { 0,1} with open sets { ∅, { 1}, { 0,1}} the specialization order is the natural one (0 ≤ 0, 0 ≤ 1, and 1 ≤ 1). If p, q are elements of Spec(R) (the spectrum of a commutative ring R) then p ≤ q if and only if q ⊆ p (as prime ideals). Thus the closed points of Spec(R) are precisely the maximal ideals.
Important properties
As suggested by the name, the specialization preorder is a preorder, i.e. it is reflexive and transitive.