The specialization order is often considered in applications in computer science, where T0 spaces occur in denotational semantics.
What is agreed[citation needed] is that if (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.
Both definitions have intuitive justifications: in the case of the former, we have 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 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, 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.
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.
The closed points of a topological space X are precisely the minimal elements of X with respect to ≤.
The equivalence relation determined by the specialization preorder is just that of topological indistinguishability.
On the other hand, the symmetry of the specialization preorder is equivalent to the R0 separation axiom: x ≤ y if and only if x and y are topologically indistinguishable.
Hence, the specialization order is of little interest for T1 topologies, especially for all Hausdorff spaces.
This functor has a left adjoint, which places the Alexandrov topology on a preordered set.
Their relationship to the specialization order is more subtle: For any sober space X with specialization order ≤, we have One may describe the second property by saying that open sets are inaccessible by directed suprema.
The specialization order yields a tool to obtain a preorder from every topology.