Alexandrov topology
Encyclopedia
In topology
, an Alexandrov space (or Alexandrov-discrete space) is a topological space
in which the intersection
of any family of open set
s is open. It is an axiom of topology that the intersection of any finite family of open sets is open. In an Alexandrov space the finite restriction is strengthened.
Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any preorder
≤ on a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the upper set
s with respect to ≤. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X.
Alexandrov spaces are also called finitely generated spaces since their topology is uniquely determined by
the family of all finite subspaces. Alexandrov spaces can be viewed as a generalization of finite topological space
s.
Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician Pavel Alexandrov who first investigated them.
we can define an Alexandrov topology 
on X by choosing the open sets to be the upper set
s:
We thus obtain a topological space
.
The corresponding closed sets are the lower sets:
We thus obtain a preordered set W(X) = <X, ≤>.
Moreover for every Alexandrov space X, we have T(W(X)) = X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.
However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with a finer topology than that of X (i.e. it will have more open sets).
between two preordered sets (i.e. a function
between the underlying sets such that x≤y in X implies f(x)≤f(y) in Y), let
be the same map as f considered as a map between the corresponding Alexandrov spaces. Then
is a continuous map.
Conversely given a continuous map
between two topological spaces, let
be the same map as f considered as a map between the corresponding preordered sets. Then
is a monotone function.
Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.
Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets. (To see this consider a non-Alexandrov space X and consider the identity map
and maps
. Let Top denote the category of topological spaces
and continuous maps; and let Pro denote the category of preordered sets
and monotone functions. Then
are concrete functors over Set which are left and right adjoints
respectively.
Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions
are inverse concrete isomorphisms over Set.
Alx is in fact a coreflective subcategory of Top with coreflector T◦W : Top→Alx. This means that given a topological space X, the identity map
is continuous and for every continuous map
where Y is an Alexandrov space, the composition
is continuous.
of T(X) are given by:
Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X, this construction is a special case of the construction of a modal algebra
from a modal frame
i.e. a set with a single binary relation
. (The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i.e. a set with relations defined on it.) The class of modal algebras that we obtain in the case of a preordered set is the class of interior algebra
s—the algebraic abstractions of topological spaces.
s later came to be used for topological spaces in which every subset is open and the original concept lay forgotten. With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation
was applied to general topology and the name finitely generated spaces was adopted for them. Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from denotational semantics
and domain theory
in computer science
.
Michael C. McCord had observed that there was a duality between partially ordered set
s and spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced. P. Johnstone referred to such topologies as Alexandrov topologies. F. G. Arenas independently proposed this name for the general version of these topologies. McCord also showed that these spaces are weak homotopy equivalent to the order complex of the corresponding partially ordered set.
It was also a well known result in the field of modal logic
that a duality exists between finite topological spaces and preorders on finite sets (the finite modal frame
s for the modal logic S4). C. Naturman extended these results to a duality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as the interior and closure algebraic
characterizations.
A systematic investigation of these spaces from the point of view of general topology which had been neglected since the original paper by Alexandrov, was taken up by F.G. Arenas.
Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990s began investigating the Alexandrov topology corresponding to causal sets
which arise from a preorder defined on spacetime
modeling causality
.
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...
, an Alexandrov space (or Alexandrov-discrete space) is 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...
in which 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 any family of open set
Open 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...
s is open. It is an axiom of topology that the intersection of any finite family of open sets is open. In an Alexandrov space the finite restriction is strengthened.
Alexandrov topologies are uniquely determined by their specialization preorders. Indeed, given any 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 a set X, there is a unique Alexandrov topology on X for which the specialization preorder is ≤. The open sets are just the 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....
s with respect to ≤. Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X.
Alexandrov spaces are also called finitely generated spaces since their topology is uniquely determined by
Coherent topology
In topology, a coherent topology is one that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces.-Definition:...
the family of all finite subspaces. Alexandrov spaces can be viewed as a generalization of finite topological space
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space for which there are only finitely many points....
s.
Characterizations of Alexandrov topologies
Alexandrov topologies have numerous characterizations. Let X = <X, T> be a topological space. Then the following are equivalent:- Open and closed set characterizations:
- Open set characterization. An arbitrary intersection of open sets in X is open.
- Closed set characterization. An arbitrary union of closed sets in X is closed.
- Neighbourhood characterizations:
- Smallest neighbourhood characterization. Every point of X has a smallest neighbourhood.
- Neighbourhood filter characterization. The neighbourhood filter of every point in X is closed under arbitrary intersections.
- Interior and closure algebraic characterizations:
- Interior operator characterization. The interior operator of X distributes over arbitrary intersections of subsets.
- Closure operator characterization. The closure operatorClosure operatorIn mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
of X distributes over arbitrary unions of subsets.
- Preorder characterizations:
- Specialization preorder characterization. T is the finest topology consistent with the specialization preorder of X i.e. the finest topology giving the preorderPreorderIn mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
≤ satisfying x ≤ y if and only if x is in the closure of {y} in X. - Open up-set characterization. There is a preorder ≤ such that the open sets of X are precisely those that are upwardly closedUpper setIn 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....
i.e. if x is in the set and x ≤ y then y is in the set. (This preorder will be precisely the specialization preorder.) - Closed down-set characterization. There is a preorder ≤ such that the closed sets of X are precisely those that are downwardly closed i.e. if x is in the set and y ≤ x then y is in the set. (This preorder will be precisely the specialization preorder.)
- Upward interior characterization. A point x lies in the interior of a subset S of X if and only if there is a point y in S such that y ≤ x where ≤ is the specialization preorder i.e. y lies in the closure of {x}.
- Downward closure characterization. A point x lies in the closure of a subset S of X if and only if there is a point y in S such that x ≤ y where ≤ is the specialization preorder i.e. x lies in the closure of {y}.
- Specialization preorder characterization. T is the finest topology consistent with the specialization preorder of X i.e. the finest topology giving the preorder
- Finite generation and category theoretic characterizations:
- Finite closure characterization. A point x lies within the closure of a subset S of X if and only if there is a finite subset F of S such that x lies in the closure of F.
- Finite subspace characterization. T is coherentCoherent topologyIn topology, a coherent topology is one that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces.-Definition:...
with the finite subspaces of X. - Finite inclusion map characterization. The inclusion maps fi : Xi → X of the finite subspaces of X form a final sink.
- Finite generation characterization. X is finitely generated i.e. it is in the final hull of the finite spaces. (This means that there is a final sink fi : Xi → X where each Xi is a finite topological space.)
Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov spaces and their topology T is called the Alexandrov topology, named after the Russian mathematician Pavel Alexandrov who first investigated them.
The Alexandrov topology on a preordered set
Given a preordered set we can define an Alexandrov topology on X by choosing the open sets to be the upper setUpper 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....
s:
We thus obtain a topological space
.The corresponding closed sets are the lower sets:
The specialization preorder on a topological space
Given a topological space X = <X, T> the specialization preorder on X is defined by:- x≤y if and only if x is in the closure of {y}.
We thus obtain a preordered set W(X) = <X, ≤>.
Equivalence between preorders and Alexandrov topologies
For every preordered set X = <X, ≤> we always have W(T(X)) = X, i.e. the preorder of X is recovered from the topological space T(X) as the specialization preorder.Moreover for every Alexandrov space X, we have T(W(X)) = X, i.e. the Alexandrov topology of X is recovered as the topology induced by the specialization preorder.
However for a topological space in general we do not have T(W(X)) = X. Rather T(W(X)) will be the set X with a finer topology than that of X (i.e. it will have more open sets).
Equivalence between monotony and continuity
Given a monotone function- f : X→Y
between two preordered sets (i.e. a function
- f : X→Y
between the underlying sets such that x≤y in X implies f(x)≤f(y) in Y), let
- T(f) : T(X)→T(Y)
be the same map as f considered as a map between the corresponding Alexandrov spaces. Then
- T(f) : T(X)→T(Y)
is a continuous map.
Conversely given a continuous map
- f : X→Y
between two topological spaces, let
- W(f) : W(X)→W(Y)
be the same map as f considered as a map between the corresponding preordered sets. Then
- W(f) : W(X)→W(Y)
is a monotone function.
Thus a map between two preordered sets is monotone if and only if it is a continuous map between the corresponding Alexandrov spaces. Conversely a map between two Alexandrov spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.
Notice however that in the case of topologies other than the Alexandrov topology, we can have a map between two topological spaces that is not continuous but which is nevertheless still a monotone function between the corresponding preordered sets. (To see this consider a non-Alexandrov space X and consider the identity map
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
- i : X→T(W(X)).)
Category theoretic description of the duality
Let Set denote the category of setsCategory of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
and maps
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
. Let Top denote 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...
and continuous maps; and let Pro denote the category of preordered sets
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...
and monotone functions. Then
- T : Pro→Top and
- W : Top→Pro
are concrete functors over Set which are left and right adjoints
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
respectively.
Let Alx denote the full subcategory of Top consisting of the Alexandrov spaces. Then the restrictions
- T : Pro→Alx and
- W : Alx→Pro
are inverse concrete isomorphisms over Set.
Alx is in fact a coreflective subcategory of Top with coreflector T◦W : Top→Alx. This means that given a topological space X, the identity map
- i : T(W(X))→X
is continuous and for every continuous map
- f : Y→X
where Y is an Alexandrov space, the composition
- i -1◦f : Y→T(W(X))
is continuous.
Relationship to the construction of modal algebras from modal frames
Given a preordered set X, the interior operator and closure operatorClosure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
of T(X) are given by:
- Int(S) = { x ∈ X : for all y ∈ X, x≤y implies y ∈ S }, for all S ⊆ X
- Cl(S) = { x ∈ X : there exists a y ∈ S with x≤y } for all S ⊆ X
Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X, this construction is a special case of the construction of a modal algebra
Modal algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that*\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra,...
from a modal frame
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
i.e. a set with a single binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
. (The latter construction is itself a special case of a more general construction of a complex algebra from a relational structure i.e. a set with relations defined on it.) The class of modal algebras that we obtain in the case of a preordered set is the class of interior algebra
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic...
s—the algebraic abstractions of topological spaces.
History
Alexandrov spaces were first introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where he provided the characterizations in terms of sets and neighbourhoods. The name discrete spaceDiscrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
s later came to be used for topological spaces in which every subset is open and the original concept lay forgotten. With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation
Finitely generated
In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
was applied to general topology and the name finitely generated spaces was adopted for them. Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from 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...
and 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...
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...
.
Michael C. McCord had observed that there was a duality between partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
s and spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced. P. Johnstone referred to such topologies as Alexandrov topologies. F. G. Arenas independently proposed this name for the general version of these topologies. McCord also showed that these spaces are weak homotopy equivalent to the order complex of the corresponding partially ordered set.
It was also a well known result in the field of modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
that a duality exists between finite topological spaces and preorders on finite sets (the finite modal frame
Modal frame
In music a melodic mode or modal frame is one of, "a number of types permeating and unifying African, European, and American song" and melody. "Mode" and "frame" are used in this context interchangeably. Melodic modes allow melodies which are not chord-based or determined by the harmony but...
s for the modal logic S4). C. Naturman extended these results to a duality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as the interior and closure algebraic
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic...
characterizations.
A systematic investigation of these spaces from the point of view of general topology which had been neglected since the original paper by Alexandrov, was taken up by F.G. Arenas.
Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990s began investigating the Alexandrov topology corresponding to causal sets
Causal sets
The causal sets programme is an approach to quantum gravity. Its founding principle is that spacetime is fundamentally discrete and that the spacetime events are related by a partial order...
which arise from a preorder defined on spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
modeling causality
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
.