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| In [[topology]], an '''Alexandrov space''' (or '''Alexandrov-discrete space''') is a [[topological space]] in which the [[intersection (set theory)|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 dropped.
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| Alexandrov topologies are uniquely determined by their [[specialization preorder]]s. Indeed, given any [[preorder]] ≤ on a [[Set (mathematics)|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''.
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| Alexandrov spaces are also called '''finitely generated spaces''' since their topology is uniquely [[coherent topology|determined by]] the family of all finite subspaces. Alexandrov spaces can be viewed as a generalization of [[finite topological space]]s.
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| == Characterizations of Alexandrov topologies ==
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| Alexandrov topologies have numerous characterizations. Let '''''X''''' = <''X'', ''T''> be a topological space. Then the following are equivalent:
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|
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| *'''Open and closed set characterizations:'''
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| ** '''Open set.''' An arbitrary intersection of open sets in '''''X''''' is open.
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| ** '''Closed set.''' An arbitrary union of closed sets in '''''X''''' is closed.
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| *'''Neighbourhood characterizations:'''
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| ** '''Smallest neighbourhood.''' Every point of '''''X''''' has a smallest [[neighbourhood (topology)|neighbourhood]].
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| ** '''Neighbourhood filter.''' The [[neighbourhood filter]] of every point in '''''X''''' is closed under arbitrary intersections.
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| *'''Interior and closure algebraic characterizations:'''
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| ** '''Interior operator.''' The [[interior operator]] of '''''X''''' distributes over arbitrary intersections of subsets.
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| ** '''Closure operator.''' The [[closure operator]] of '''''X''''' distributes over arbitrary unions of subsets.
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| *'''Preorder characterizations:'''
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| ** '''Specialization preorder.''' ''T'' is the [[finest topology]] consistent with the [[specialization preorder]] of '''''X''''' i.e. the finest topology giving the [[preorder]] ≤ satisfying ''x'' ≤ ''y'' if and only if ''x'' is in the closure of {''y''} in '''''X'''''.
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| ** '''Open up-set.''' There is a preorder ≤ such that the open sets of '''''X''''' are precisely those that are [[upper set|upwardly closed]] 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.)
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| ** '''Closed down-set.''' 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.)
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| ** '''Upward interior.''' 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''}.
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| ** '''Downward closure.''' 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''}.
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| *'''Finite generation and category theoretic characterizations:'''
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| ** '''Finite closure.''' 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''.
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| ** '''Finite subspace.''' ''T'' is [[coherent topology|coherent]] with the finite subspaces of '''''X'''''.
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| ** '''Finite inclusion map.''' The inclusion maps ''f''<sub>''i''</sub> : '''''X'''''<sub>''i''</sub> → '''''X''''' of the finite subspaces of '''''X''''' form a [[final sink]].
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| ** '''Finite generation.''' '''''X''''' is finitely generated i.e. it is in the [[final hull]] of the finite spaces. (This means that there is a final sink ''f''<sub>''i''</sub> : '''''X'''''<sub>''i''</sub> → '''''X''''' where each '''''X'''''<sub>''i''</sub> is a finite topological space.)
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| 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.
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| == Duality with preordered sets ==
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| === The Alexandrov topology on a preordered set ===
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| Given a [[preordered set]] <math> \mathbf{X} = \langle X, \le\rangle</math> we can define an Alexandrov topology <math>\tau</math> on ''X'' by choosing the open sets to be the [[upper set]]s:
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| :<math>\tau = \{\, G \subseteq X : \forall x,y\in X\ \ x\in G\ \land\ x\le y\ \rightarrow\ y \in G,\}</math>
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| We thus obtain a topological space <math>\mathbf{T}(\mathbf{X}) = \langle X, \tau\rangle</math>.
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| The corresponding closed sets are the [[lower set]]s:
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| ::<math>\{\, S \subseteq X : \forall x,y\in X\ \ x\in S\ \land\ y\le x\ \rightarrow\ y \in S,\}</math>
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| === The specialization preorder on a topological space === | |
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| Given a topological space '''''X''''' = <''X'', ''T''> the [[specialization preorder]] on ''X'' is defined by:
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| : ''x''≤''y'' if and only if ''x'' is in the closure of {''y''}.
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| We thus obtain a preordered set '''''W'''''('''''X''''') = <''X'', ≤>.
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| === Equivalence between preorders and Alexandrov topologies ===
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| 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.
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| 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.
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| 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).
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| === Equivalence between monotony and continuity ===
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| Given a [[monotone function]]
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| :''f'' : '''''X'''''→'''''Y'''''
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| between two preordered sets (i.e. a function
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| :''f'' : ''X''→''Y''
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| between the underlying sets such that ''x''≤''y'' in '''''X''''' implies ''f''(''x'')≤''f''(''y'') in '''''Y'''''), let
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| :'''''T'''''(''f'') : '''''T'''''('''''X''''')→'''''T'''''('''''Y''''')
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| be the same map as ''f'' considered as a map between the corresponding Alexandrov spaces. Then
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| :'''''T'''''(''f'') : '''''T'''''('''''X''''')→'''''T'''''('''''Y''''')
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| is a [[continuous map (topology)|continuous map]].
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| Conversely given a continuous map
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| :''f'' : '''''X'''''→'''''Y'''''
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| between two topological spaces, let
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| :'''''W'''''(''f'') : '''''W'''''('''''X''''')→'''''W'''''('''''Y''''')
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| be the same map as ''f'' considered as a map between the corresponding preordered sets. Then
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| :'''''W'''''(''f'') : '''''W'''''('''''X''''')→'''''W'''''('''''Y''''')
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| is a monotone function.
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| 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.
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| 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 function|identity map]]
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| :''i'' : '''''X'''''→'''''T'''''('''''W'''''('''''X''''')).)
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| ===Category theoretic description of the duality===
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| Let '''Set''' denote the [[category of sets]] and [[map (mathematics)|maps]]. Let '''Top''' denote the [[category of topological spaces]] and [[continuity (topology)|continuous maps]]; and let '''Pro''' denote the category of [[preorder|preordered sets]] and [[monotone function]]s. Then
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| :'''''T''''' : '''Pro'''→'''Top''' and
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| :'''''W''''' : '''Top'''→'''Pro'''
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| are [[concrete functor]]s over '''Set''' which are [[adjoint functors|left and right adjoints]] respectively.
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| Let '''Alx''' denote the [[full subcategory]] of '''Top''' consisting of the Alexandrov spaces. Then the restrictions
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| :'''''T''''' : '''Pro'''→'''Alx''' and
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| :'''''W''''' : '''Alx'''→'''Pro'''
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| are inverse [[concrete functor|concrete isomorphisms]] over '''Set'''.
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| '''Alx''' is in fact a [[coreflective subcategory|bico-reflective subcategory]] of '''Top''' with bico-reflector '''''T'''''◦'''''W''''' : '''Top'''→'''Alx'''. This means that given a topological space '''''X''''', the identity map
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| :''i'' : '''''T'''''('''''W'''''('''''X'''''))→'''''X'''''
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| is continuous and for every continuous map | |
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| :''f'' : '''''Y'''''→'''''X'''''
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| where '''''Y''''' is an Alexandrov space, the composition
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| :''i'' <sup>-1</sup>◦''f'' : '''''Y'''''→'''''T'''''('''''W'''''('''''X'''''))
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| is continuous.
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| === Relationship to the construction of modal algebras from modal frames ===
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| Given a preordered set '''''X''''', the [[interior operator]] and [[closure operator]] of '''''T'''''('''''X''''') are given by:
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| :'''Int'''(''S'') = { ''x'' ∈ X : for all ''y'' ∈ X, ''x''≤''y'' implies ''y'' ∈ S }, for all ''S'' ⊆ ''X''
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| :'''Cl'''(''S'') = { ''x'' ∈ X : there exists a ''y'' ∈ S with ''x''≤''y'' } for all ''S'' ⊆ ''X''
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| Considering the interior operator and closure operator to be modal operators on the [[power set]] [[Boolean algebra (structure)|Boolean algebra]] of ''X'', this construction is a special case of the construction of a [[modal algebra]] from a [[Kripke semantics|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.
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| == History ==
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| 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.<ref name="Ale37">{{cite journal |last=Alexandroff |first=P. |title=Diskrete Räume |journal=Mat. Sb. (N.S.) |volume=2 |issue= |year=1937 |pages=501–518 |doi= |url=http://mi.mathnet.ru/rus/msb/v44/i3/p501 |language=German }}</ref> The name [[discrete space]]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 [[Finitely generated object|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]].
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| In 1966 Michael C. McCord and A. K. Steiner each independently observed a duality between [[partially ordered set]]s and spaces which were precisely the [[Kolomogorov space|T<sub>0</sub>]] versions of the spaces that Alexandrov had introduced.<ref name="McC66">{{cite journal |last=McCord |first=M. C. |title=Singular homology and homotopy groups of finite topological spaces |journal=[[Duke Mathematical Journal]] |volume=33 |issue=3 |year=1966 |pages=465–474 |doi=10.1215/S0012-7094-66-03352-7 }}</ref><ref name="Ste66">{{cite journal |last=Steiner |first=A. K. |title=The Lattice of Topologies: Structure and Complementation |journal=[[Transactions of the American Mathematical Society]] |volume=122 |issue=2 |year=1966 |pages=379–398 |doi=10.2307/1994555 | ISSN=00029947 }}</ref> P. Johnstone referred to such topologies as '''Alexandrov topologies'''.<ref name="Joh82">{{cite book |last=Johnstone |first=P. T. |title=Stone spaces |location=New York |publisher=Cambridge University Press |year=1986 |edition=1st paperback |isbn=0-521-33779-8 }}</ref> F. G. Arenas independently proposed this name for the general version of these topologies.<ref name="Are99">{{cite journal |last=Arenas |first=F. G. |title=Alexandroff spaces |journal=Acta Math. Univ. Comenianae |volume=68 |issue=1 |year=1999 |pages=17–25 |doi= |url=http://www.emis.ams.org/journals/AMUC/_vol-68/_no_1/_arenas/arenas.pdf }}</ref> McCord also showed that these spaces are [[weak homotopy equivalence|weak homotopy equivalent]] to the [[order complex]] of the corresponding partially ordered set. Steiner demonstrated that the duality is a [[Covariance_and_contravariance_of_functors|contravariant]] [[Lattice (order)|lattice]] isomorphism preserving [[Complete_lattice|arbitrary meets and joins]] as well as complementation.
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| 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 algebra|interior and closure algebraic]] characterizations.<ref name="Nat91">{{cite book |last=Naturman |first=C. A. |title=Interior Algebras and Topology |publisher=Ph.D. thesis, University of Cape Town Department of Mathematics |year=1991 }}</ref>
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| 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.<ref name="Are99" />
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| 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]].
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| == See also ==
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| * [[P-space|''P''-space]], a space satisfying the weaker condition that countable intersections of open sets are open
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| == References ==
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| {{Reflist}}
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| {{DEFAULTSORT:Alexandrov Topology}}
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| [[Category:Properties of topological spaces]]
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| [[Category:Order theory]]
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| [[Category:Closure operators]]
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