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.

Alexandrov topologies have numerous characterizations. In a topological space ${\displaystyle X}$, the following conditions are equivalent:

An Alexandrov topology is canonically associated to a preordered set by taking the open sets to be the upper sets. Conversely, the preordered set can be recovered from the Alexandrov topology as its specialization preorder. (We use the convention that the specialization preorder is defined by ${\displaystyle x\leq y}$ whenever ${\displaystyle x\in \operatorname {cl} \{y\},}$ that is, when every open set that contains ${\displaystyle x}$ also contains ${\displaystyle y}$, to match our convention that the open sets in the Alexandrov topology are the upper sets rather than the lower sets; the opposite convention also exists.)

The following dictionary holds between order-theoretic notions and topological notions:

From the point of view of category theory, let Top denote the category of topological spaces consisting of topological spaces with continuous maps as morphisms. Let Alex denote its full subcategory consisting of Alexandrov-discrete spaces. Let Preord denote the category of preordered sets consisting of preordered sets with order preserving functions as morphisms. The correspondence above is an isomorphism of categories between Alex and PreOrd.

Furthermore, the functor ${\displaystyle A:\mathbf {PreOrd} \to \mathbf {Top} }$ that sends a preordered set to its associated Alexandrov-discrete space is fully faithful and left adjoint to the specialization preorder functor ${\displaystyle S:\mathbf {Top} \to \mathbf {PreOrd} }$, making Alex a coreflective subcategory of Top. Moreover, the reflection morphisms ${\displaystyle A(S(X))\to X}$, whose underlying maps are the identities (but with different topologies at the source and target), are bijective continuous maps, thus bimorphisms.