Alexandrov topologies are uniquely determined by their specialization preorders.
Thus, Alexandrov topologies on X are in one-to-one correspondence with preorders on X. Alexandrov-discrete spaces are also called finitely generated spaces because their topology is uniquely determined by the family of all finite subspaces.
Because 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.
Given a monotone function between two preordered sets (i.e. a function between the underlying sets such that x ≤ y in X implies f(x) ≤ f(y) in Y), let be the same map as f considered as a map between the corresponding Alexandrov spaces.
Conversely a map between two Alexandrov-discrete spaces is continuous if and only if it is a monotone function between the corresponding preordered sets.
Let Alx denote the full subcategory of Top consisting of the Alexandrov-discrete spaces.
Alx is in fact a bico-reflective subcategory of Top with bico-reflector T◦W : Top→Alx.
Considering the interior operator and closure operator to be modal operators on the power set Boolean algebra of X, this construction is a special case of the construction of a modal algebra from a modal frame i.e. from a set with a single binary relation.
On the other hand, Alexandrov spaces played a relevant role in Øystein Ore pioneering studies on closure systems and their relationships with lattice theory and topology.
[7] With the advancement of categorical topology in the 1980s, Alexandrov spaces were rediscovered when the concept of finite generation was applied to general topology and the name finitely generated spaces was adopted for them.
Alexandrov spaces were also rediscovered around the same time in the context of topologies resulting from denotational semantics and domain theory in computer science.
In 1966 Michael C. McCord and A. K. Steiner each independently observed an equivalence between partially ordered sets and spaces that were precisely the T0 versions of the spaces that Alexandrov had introduced.
[10] F. G. Arenas independently proposed this name for the general version of these topologies.
Steiner demonstrated that the equivalence is a contravariant lattice isomorphism preserving arbitrary meets and joins as well as complementation.
A. Grzegorczyk observed that this extended to a equivalence between what he referred to as totally distributive spaces and preorders.
C. Naturman observed that these spaces were the Alexandrov-discrete spaces and extended the result to a category-theoretic equivalence between the category of Alexandrov-discrete spaces and (open) continuous maps, and the category of preorders and (bounded) monotone maps, providing the preorder characterizations as well as the interior and closure algebraic characterizations.
[12] A systematic investigation of these spaces from the point of view of general topology, which had been neglected since the original paper by Alexandrov was taken up by F. G.