In the branch ofmathematics known astopology, thespecialization (orcanonical)preorder is a naturalpreorder on the set of the points of atopological space. For most spaces that are considered in practice, namely for all those that satisfy theT0separation axiom, this preorder is even apartial order (called thespecialization order). On the other hand, forT1 spaces the order becomes trivial and is of little interest.
The specialization order is often considered in applications incomputer science, where T0 spaces occur indenotational semantics. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done inorder theory.
Consider any topological spaceX. Thespecialization preorder ≤ onX relates two points ofX when one lies in theclosure of the other. However, various authors disagree on which 'direction' the order should go. What is agreed[citation needed] is that if
(where cl{y} denotes the closure of thesingleton set {y}, i.e. theintersection of allclosed sets containing {y}), we say thatx is aspecialization ofy and thaty is ageneralization ofx; this is commonly writteny ⤳ x.
Unfortunately, the property "x is a specialization ofy" is alternatively written as "x ≤y" and as "y ≤x" by various authors (see, respectively,[1] and[2]).
Both definitions have intuitive justifications: in the case of the former, we have
However, in the case where our spaceX is theprime spectrum Spec(R) of acommutative ringR (which is the motivational situation in applications related toalgebraic 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 ofy" be written asx ≤y. We then see,
These restatements help to explain why one speaks of a "specialization":y is more general thanx, since it is contained in more open sets. This is particularly intuitive if one views closed sets as properties that a pointx 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 isconsistent with the classical logical notions ofgenus andspecies; and also with the traditional use ofgeneric points inalgebraic 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 invaluation theory.
The intuition of upper elements being more specific is typically found indomain theory, a branch of order theory that has ample applications in computer science.
LetX be a topological space and let ≤ be the specialization preorder onX. Everyopen set is anupper set with respect to ≤ and everyclosed set is alower set. The converses are not generally true. In fact, a topological space is anAlexandrov-discrete space if and only if every upper set is also open (or equivalently every lower set is also closed).
LetA be a subset ofX. The smallest upper set containingA is denoted ↑Aand the smallest lower set containingA is denoted ↓A. In caseA = {x} is a singleton one uses the notation ↑x and ↓x. Forx ∈X one has:
The lower set ↓x is always closed; however, the upper set ↑x need not be open or closed. The closed points of a topological spaceX are precisely theminimal elements ofX with respect to ≤.
As suggested by the name, the specialization preorder is a preorder, i.e. it isreflexive andtransitive.
Theequivalence relation determined by the specialization preorder is just that oftopological indistinguishability. That is,x andy are topologically indistinguishable if and only ifx ≤y andy ≤x. Therefore, theantisymmetry of ≤ is precisely the T0 separation axiom: ifx andy are indistinguishable thenx =y. In this case it is justified to speak of thespecialization order.
On the other hand, thesymmetry of the specialization preorder is equivalent to theR0 separation axiom:x ≤y if and only ifx andy are topologically indistinguishable. It follows that if the underlying topology is T1, then the specialization order is discrete, i.e. one hasx ≤y if and only ifx =y. Hence, the specialization order is of little interest for T1 topologies, especially for allHausdorff spaces.
Anycontinuous function between two topological spaces ismonotone with respect to the specialization preorders of these spaces: implies The converse, however, is not true in general. In the language ofcategory theory, we then have afunctor from thecategory of topological spaces to thecategory of preordered sets that assigns a topological space its specialization preorder. This functor has aleft adjoint, which places theAlexandrov topology on a preordered set.
There are spaces that are more specific than T0 spaces for which this order is interesting: thesober spaces. Their relationship to the specialization order is more subtle:
For any sober spaceX with specialization order ≤, we have
One may describe the second property by saying that open sets areinaccessible by directed suprema. A topology isorder 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 ≤.
The specialization order yields a tool to obtain a preorder from every topology. It is natural to ask for the converse too: Is every preorder obtained as a specialization preorder of some topology?
Indeed, the answer to this question is positive and there are in general many topologies on a setX that induce a given order ≤ as their specialization order. TheAlexandroff topology of the order ≤ plays a special role: it is the finest topology that induces ≤. The other extreme, the coarsest topology that induces ≤, is theupper topology, the least topology within which all complements of sets ↓x (for somex inX) are open.
There are also interesting topologies in between these two extremes. The finest sober topology that is order consistent in the above sense for a given order ≤ is theScott topology. The upper topology however is still the coarsest sober order-consistent topology. In fact, its open sets are even inaccessible byany suprema. Hence anysober space with specialization order ≤ is finer than the upper topology and coarser than the Scott topology. Yet, such a space may fail to exist, that is, there exist partial orders for which there is no sober order-consistent topology. Especially, the Scott topology is not necessarily sober.