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 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 if and only if cl{x} is a subset of cl{y},

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


 * x ≤ y if and only if 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.

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 space if and only if every upper set is open (or every closed set is lower).

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, which is indeed easy to see.

The equivalence relation determined by the specialization preorder is just that of topological indistinguishability. That is, x and y are topologically indistinguishable if and only if x ≤ y and  y ≤ x. Therefore, the antisymmetry of ≤ is precisely the T0 separation axiom: if x and y are indistinguishable then x = y. In this case it is justified to speak of the specialization order.

On the other hand, the symmetry of specialization preorder is equivalent to the R0 separation axiom: x ≤ y if and only if x and y are topologically indistinguishable. It follows that if the underlying topology is T1, then the specialization order is discrete, i.e. one has x ≤ y if and only if x = y. Hence, the specialization order is of little interest for T1 topologies, especially for all Hausdorff spaces.

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. In the language of category theory, we then have a functor from the category of topological spaces to the category of preordered sets which assigns a topological space its specialization preorder. This functor has a left adjoint which places the Alexandrov topology on a preordered set.

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 order consistent with respect to a certain order ≤ if it induces ≤ as its specialization order and it has the above property of inaccessibility with respect to (existing) suprema of directed sets in ≤.

Topologies on orders
The specialization order yields a tool to obtain a partial order from every topology. It is natural to ask for the converse too: Is every partial order obtained as a specialization order of some topology?

Indeed, the answer to this question is positive and there are in general many topologies on a set X which induce a given order ≤ as their specialization order. The Alexandroff 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 Scott topology. The upper topology however is still the coarsest order consistent topology. In fact its open sets are even inaccessible by any suprema. Hence any sober space with specialization order ≤ is finer than the upper topology and coarser than the Scott topology. Yet, such a space may fail to exist. Especially, the Scott topology is not necessarily sober.