Alexandrov topology

In topology, an Alexandrov topology is a topology 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 Alexandrov topologies the finite restriction is dropped. In topology, an Alexandrov topology is a topology 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 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 inverse images 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 = <X, T> be a topological space. Then the following are equivalent: Topological spaces satisfying the above equivalent characterizations are called finitely generated spaces or Alexandrov-discrete spaces and their topology T is called an Alexandrov topology. Given a preordered set X = ⟨ X , ≤ ⟩ {displaystyle mathbf {X} =langle X,leq angle } we can define an Alexandrov topology τ {displaystyle au } on X by choosing the open sets to be the upper sets: