Alexandrov Topology

Alexandrov Topology

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.