# Specialization (pre)order

In the branch of

mathematics known astopology , the**specialization**(or**canonical**)**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 the T_{0}separation axiom , this preorder is even apartial order (called the**specialization order**). On the other hand, for T_{1}spaces the order becomes trivial and is of little interest.The specialization order is often considered in applications in

computer science , where T_{0}spaces occur indenotational semantics . The specialization order is also important for identifying suitable topologies on partially ordered sets, as it is done inorder 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 allclosed set s 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 set s 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 ofgenus andspecies ; and also with the traditional use ofgeneric point s inalgebraic geometry . Specialization as an idea is applied also invaluation 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 anupper set with respect to ≤ and everyclosed set is alower set . The converses are not generally true. In fact, a topological space is anAlexandrov 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 element s 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 acommutative ring "R") then "p" ≤ "q" if and only if "q" ⊆ "p" (asprime ideal s). Thus the closed points of Spec("R") are precisely themaximal ideal s.**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 T_{0}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 R

_{0}separation axiom: "x" ≤ "y" if and only if "x" and "y" are topologically indistinguishable. It follows that if the underlying topology is T_{1}, 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 T_{1}topologies, especially for allHausdorff space s.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 afunctor from thecategory of topological spaces to thecategory of preordered sets which 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 T

_{0}spaces for which this order is interesting: thesober space s. Their relationship to the specialization order is more subtle:For any sober space "X" with specialization order ≤, we have

* ("X", ≤) is adirected complete partial order , i.e. every directed subset "S" of ("X", ≤) has asupremum sup "S",

* for every directed subset "S" of ("X", ≤) and every open set "O", if sup "S" is in "O", then "S" and "O" havenon-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 theupper 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.**References*** M.M. Bonsangue, "Topological Duality in Semantics", volume 8 of Electronic Notes in Theoretical Computer Science, 1998. Revised version of author's Ph.D. thesis. Available [

*http://www1.elsevier.com/gej-ng/31/29/23/52/23/show/Products/notes/index.htt online*] , see especially [*http://www1.elsevier.com/gej-ng/31/29/23/52/23/53/tcs8007.ps Chapter 5*] , that explains the motivations from the viewpoint of denotational semantics in computer science. See also the authors [*http://www.liacs.nl/~marcello/ homepage*] .

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