Alexandrov topology

In general topology, an Alexandrov topology is a topology in which the intersection of an arbitrary family of open sets is open (while the definition of a topology only requires this for a finite family). Equivalently, an Alexandrov topology is one whose open sets are the upper sets for some preorder on the space.

Spaces with an Alexandrov topology are also known as Alexandrov-discrete spaces or finitely generated spaces. The latter name stems from the fact that their topology is uniquely determined by the family of all finite subspaces. This makes them a generalization of finite topological spaces.

Alexandrov-discrete spaces are named after the Russian topologist Pavel Alexandrov. They should not be confused with Alexandrov spaces from Riemannian geometry introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov.