Editing Specialization (pre)order (section)

== 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.