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Concept
Alexandrov topology
Summary
In topology, an Alexandrov topology is a topology in which the intersection of every family of open sets is open. It is an axiom of topology that the intersection of every finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped.
A set together with an Alexandrov topology is known as an Alexandrov-discrete space or finitely generated space.
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-discrete spaces are also called finitely generated spaces since their topology is uniquely determined by the family of all finite subspaces. Alexandrov-discrete spaces can thus be viewed as a generalization of finite topological spaces.
Due to the fact that commute with arbitrary unions and intersections, the property of being an Alexandrov-discrete space is preserved under quotients.
Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with the more geometrical Alexandrov spaces introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.
Alexandrov topologies have numerous characterizations. Let X = 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.
Official source
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