Specialization order

In the branch of mathematics known as topology the specialization (or canonical) preorder defines a 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. 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 it is done in order theory.

Definition and motivation

Consider any topological space X. The specialization preorder ≤ on X is defined by setting

x ≤ y iff cl{x} is a subset of cl{y},

where cl{x} denotes the closure of the singleton set {x}. While this brief definition is convenient, it is helpful to note that the following statement is equivalent:

x ≤ y iff y is contained in all open sets that contain x.

This definition explains why one speaks of a "specialization": y is more special than x, since it is contained in more open sets. This is particularly intuitive if one views open sets as properties that a point x may or may not have. The more open sets contain a point, the more properties it 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. 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.

Important properties

As suggested by the name, the specialization preorder is a preorder, i.e. it is reflexive and transitive, which is indeed easy to see. Its antisymmetry is precisely the T₀ separation axiom: for every two points, there is an open set which contains only one of them. In this case it is justified to speak of the specialization order.

On the other hand, if the underlying topology is T₁, then the specialization order is trivial, i.e. one has x ≤ y iff x = y. Hence, the specialization order is of little interest for T₁ topologies, especially for all Hausdorff spaces.

Furthermore, any continuous function between two topological spaces is monotone with respect to the specialization preorders of these spaces. The converse, however, is not true in general.

There are spaces that are more specific than T₀ spaces for which this order is interesting: the sober spaces. Their relationship to the specialization order is more subtle:

For any sober space X with specialization order ≤, we have

• (X, ≤) is a directed complete partial order, i.e. every directed subset S of (X, ≤) has a supremum sup S,
• for every directed subset S of (X, ≤) and every open set O, if sup S is in O, then S and O have non-empty intersection.

One may describe the second property by saying that open sets are inaccessible by directed suprema. A topology is Alexandrov topology of the order ≤ plays a special role: it is the finest topology that induces ≤. The other extreme, the coarsest topology that induces ≤, is the upper topology, the least topology within which all complements of sets {y in X | y ≤ x} (for some x in X) are open.

There are also interesting topologies in between these two extremes. The finest topology that is order consistent in the above sense for a given order ≤ is the