Editing 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 space|T<sub>0</sub>]] [[separation axiom]], this preorder is even a [[partial order]] (called the '''specialization order'''). On the other hand, for [[T1 space|T<sub>1</sub> spaces]] the order becomes trivial and is of little interest.

The specialization order is often considered in applications in [[computer science]], where T<sub>0</sub> spaces occur in [[denotational semantics]]. The specialization order is also important for identifying suitable topologies on partially ordered sets, as is done in [[order theory]].

== Definition and motivation ==

Consider any topological space ''X''. The '''specialization preorder''' ≤ on ''X'' relates two points of ''X'' when one lies in the [[closure (topology)|closure]] of the other. However, various authors disagree on which 'direction' the order should go. What is agreed{{Citation needed|reason=This seems counter-intuitive, looks like specialization and generization could be mixed up. Consider the topology on the set of words in a finite alphabet where basic open sets are sets of words with given prefixes (each word defines the basic open set consisting of all the words that have this word as a prefix).|date=June 2017}} is that if

:''x'' is contained in cl{''y''},

(where cl{''y''} denotes the closure of the [[singleton set]] {''y''}, i.e. the [[intersection (set theory)|intersection]] of all [[closed set]]s containing  {''y''}), we say that ''x'' is a '''specialization''' of ''y'' and that ''y'' is a '''generalization''' of ''x''; this is commonly written ''y ⤳ x''.

Unfortunately, the property "''x'' is a specialization of ''y''" is alternatively written as "''x'' ≤ ''y''" and as "''y'' ≤ ''x''" by various authors (see, respectively, <ref>{{Citation| last = Hartshorne | first = Robin |authorlink = Robin Hartshorne| year = 1977 | title = Algebraic geometry | publisher = Springer-Verlag | publication-place = New York-Heidelberg | url = https://archive.org/details/springer_10.1007-978-1-4757-3849-0}}</ref> and <ref>{{Citation |last=Hochster |first=Melvin |authorlink = Melvin Hochster|year=1969 |title=Prime ideal structure in commutative rings |publisher=[[Trans. Amer. Math. Soc.]] |volume=142 |pages=43–60 |url=https://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0251026-X/S0002-9947-1969-0251026-X.pdf }}</ref>).

Both definitions have intuitive justifications: in the case of the former, we have

:''x'' ≤ ''y'' [[if and only if]] cl{''x''} ⊆ cl{''y''}.

However, in the case where our space ''X'' is the [[prime spectrum]] Spec(''R'') of a [[commutative ring]] ''R'' (which is the motivational situation in applications related to [[algebraic geometry]]), then under our second definition of the order, we have

:''y'' ≤ ''x'' if and only if ''y'' ⊆ ''x'' as [[prime ideals]] of the ring ''R''.

For the sake of consistency, for the remainder of this article we will take the first definition, that "''x'' is a specialization of ''y''" be written as ''x'' ≤ ''y''. We then see,

:''x'' ≤ ''y'' if and only if ''x'' is contained in all [[closed set]]s that contain ''y''.
:''x'' ≤ ''y'' if and only if ''y'' is contained in all [[open set]]s that contain ''x''.

These restatements help to explain why one speaks of a "specialization": ''y'' is more general than ''x'', since it is contained in more open sets. This is particularly intuitive if one views closed sets as properties that a point ''x'' 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 is [[consistent]] with the classical logical notions of [[genus]] and [[species]]; and also with the traditional use of [[generic point]]s in [[algebraic 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 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-discrete space]] if and only if every upper set is also open (or equivalently every lower set is also closed).

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 a ring|spectrum]] of a [[commutative ring]] ''R'') then ''p'' ≤ ''q'' if and only if ''q'' ⊆ ''p'' (as [[prime ideal]]s). Thus the closed points of Spec(''R'') are precisely the [[maximal ideal]]s.

== Important properties ==

As suggested by the name, the specialization preorder is a preorder, i.e. it is [[reflexive relation|reflexive]] and [[transitive relation|transitive]].

The [[equivalence relation]] determined by the specialization preorder is just that of [[Topologically indistinguishable|topological indistinguishability]]. That is, ''x'' and ''y'' are topologically indistinguishable if and only if  ''x'' ≤ ''y'' and  ''y'' ≤ ''x''. Therefore, the [[antisymmetric relation|antisymmetry]] of ≤ is precisely the T<sub>0</sub> 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 [[symmetric relation|symmetry]] of the specialization preorder is equivalent to the [[R0 space|R<sub>0</sub>]] separation axiom: ''x'' ≤ ''y'' if and only if ''x'' and ''y'' are topologically indistinguishable. It follows that if the underlying topology is T<sub>1</sub>, 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<sub>1</sub> topologies, especially for all [[Hausdorff space]]s.

Any [[continuity (topology)|continuous function]] <math>f</math> between two topological spaces is [[monotonic function|monotone]] with respect to the specialization preorders of these spaces: <math>x\le y</math> implies <math>f(x)\le f(y).</math>  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]] that 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 T<sub>0</sub> spaces for which this order is interesting: the [[sober space]]s. 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 set|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 (set theory)|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 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 set ''X'' that 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 ↓''x'' (for some ''x'' in ''X'') 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 the [[Scott topology]]. The upper topology however is still the coarsest sober 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, that is, there exist partial orders for which there is no sober order-consistent topology. Especially, the Scott topology is not necessarily sober.

==References==

{{reflist}}

===Further reading===

* 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://www.liacs.nl/~marcello/Papers/Book/ENTCS-8.pdf online], see especially Chapter 5, that explains the motivations from the viewpoint of denotational semantics in computer science. See also the author's [http://www.liacs.nl/~marcello/ homepage].

{{Order theory}}

[[Category:Order theory]]
[[Category:Topology]]