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In topology, an Alexandrov space (or Alexandrov-discrete 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. In an Alexandrov space the finite restriction is dropped. 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 sets 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 spaces. == 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. An arbitrary intersection of open sets in ''X'' is open. * * Closed set. An arbitrary union of closed sets in ''X'' is closed. *Neighbourhood characterizations: * * Smallest neighbourhood. Every point of ''X'' has a smallest neighbourhood. * * Neighbourhood filter. The neighbourhood filter of every point in ''X'' is closed under arbitrary intersections. *Interior and closure algebraic characterizations: * * Interior operator. The interior operator of ''X'' distributes over arbitrary intersections of subsets. * * Closure operator. The closure operator of ''X'' distributes over arbitrary unions of subsets. *Preorder characterizations: * * 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 in ''X''. * * Open up-set. There is a preorder ≤ such that the open sets of ''X'' are precisely those that are 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.) * * 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.) * * 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 . * * 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 . *Finite generation and category theoretic characterizations: * * 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''. * * Finite subspace. ''T'' is coherent with the finite subspaces of ''X''. * * Finite inclusion map. The inclusion maps ''f''''i'' : ''X''''i'' → ''X'' of the finite subspaces of ''X'' form a final sink. * * 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''''i'' : ''X''''i'' → ''X'' where each ''X''''i'' 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Alexandrov topology」の詳細全文を読む スポンサード リンク
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