Specialization (pre)order
Encyclopedia
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 separation axiom
, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces
the order becomes trivial and is of little interest.
The specialization order is often considered in applications in computer science
, where T0 spaces occur in denotational semantics
. The specialization order is also important for identifying suitable topologies on partially ordered sets, as it is done in order theory
.
where cl{x} denotes the closure
of the singleton set {x}, i.e. the intersection
of all closed set
s containing {x}. While this brief definition is convenient, it is helpful to note that the following statement is equivalent:
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 of genus
and species
; and also with the traditional use of generic point
s in algebraic geometry
. 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.
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 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:
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 elements of X with respect to ≤.
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 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 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 space
s. Their relationship to the specialization order is more subtle:
For any sober space X with specialization order ≤, we have
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 ≤.
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 the upper 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
known as topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the specialization (or canonical) preorder is a natural preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
on the set of the points of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. For most spaces that are considered in practice, namely for all those that satisfy the T0 separation axiom
Separation axiom
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms...
, this preorder is even a partial order (called the specialization order). On the other hand, for T1 spaces
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
the order becomes trivial and is of little interest.
The specialization order is often considered in applications in computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, where T0 spaces occur in denotational semantics
Denotational semantics
In computer science, denotational semantics is an approach to formalizing the meanings of programming languages by constructing mathematical objects which describe the meanings of expressions from the languages...
. The specialization order is also important for identifying suitable topologies on partially ordered sets, as it is done in order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
.
Definition and motivation
Consider any topological space X. The specialization preorder ≤ on X is defined by setting- x ≤ y if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
cl{x} is a subset of cl{y},
where cl{x} denotes the closure
Closure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of the singleton set {x}, i.e. the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of all closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
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 setOpen setThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
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 of genus
Genus
In biology, a genus is a low-level taxonomic rank used in the biological classification of living and fossil organisms, which is an example of definition by genus and differentia...
and species
Species
In biology, a species is one of the basic units of biological classification and a taxonomic rank. A species is often defined as a group of organisms capable of interbreeding and producing fertile offspring. While in many cases this definition is adequate, more precise or differing measures are...
; and also with the traditional use of generic point
Generic point
In mathematics, in the fields general topology and particularly of algebraic geometry, a generic point P of a topological space X is an algebraic way of capturing the notion of a generic property: a generic property is a property of the generic point.- Definition and motivation :A generic point of...
s in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. Specialization as an idea is applied also in valuation theory.
The intuition of upper elements being more specific is typically found in domain theory
Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
, 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 setOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
is an upper set
Upper set
In mathematics, an upper set of a partially ordered set is a subset U with the property that x is in U and x≤y imply y is in U....
with respect to ≤ and every closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
is a lower set. The converses are not generally true. In fact, a topological space is an Alexandrov 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 elements of X with respect to ≤.
Examples
- In the Sierpinski spaceSierpinski spaceIn mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
{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 spectrumSpectrum of a ringIn abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
of a commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R) then p ≤ q if and only if q ⊆ p (as prime idealPrime idealIn algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s). Thus the closed points of Spec(R) are precisely the maximal idealMaximal idealIn mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
s.
Important properties
As suggested by the name, the specialization preorder is a preorder, i.e. it is reflexiveReflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
and transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
, which is indeed easy to see.
The equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
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
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
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
Symmetric relation
In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
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 space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s.
Any continuous function between two topological spaces is monotone
Monotonic function
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
with respect to the specialization preorders of these spaces. The converse, however, is not true in general. In the language of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, we then have a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
to the category of preordered sets
Category of preordered sets
The category Ord has preordered sets as objects and monotonic functions as morphisms. This is a category because the composition of two monotonic functions is monotonic and the identity map is monotonic....
which assigns a topological space its specialization preorder. This functor has a left adjoint which places the Alexandrov topology
Alexandrov topology
In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
on a preordered set.
There are spaces that are more specific than T0 spaces for which this order is interesting: the sober space
Sober space
In mathematics, a sober space is a topological spacesuch that every irreducible closed subset of X is the closure of exactly one point of X: that is, has a unique generic point.-Properties and examples :...
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 subsetDirected setIn mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
S of (X, ≤) has a supremumSupremumIn mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
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 intersectionIntersection (set theory)In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
.
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 the upper 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.