Specialization (pre)order - Important Properties

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 ≤.