History
Alexandrov spaces were first introduced in 1937 by P. S. Alexandrov under the name discrete spaces, where he provided the characterizations in terms of sets and neighbourhoods. The name discrete spaces later came to be used for topological spaces in which every subset is open and the original concept lay forgotten. 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 a duality between partially ordered sets and spaces which were precisely the T0 versions of the spaces that Alexandrov had introduced. P. Johnstone referred to such topologies as Alexandrov topologies. F. G. Arenas independently proposed this name for the general version of these topologies. McCord also showed that these spaces are weak homotopy equivalent to the order complex of the corresponding partially ordered set. Steiner demonstrated that the duality is a contravariant lattice isomorphism preserving arbitrary meets and joins as well as complementation.
It was also a well known result in the field of modal logic that a duality exists between finite topological spaces and preorders on finite sets (the finite modal frames for the modal logic S4). C. Naturman extended these results to a duality between Alexandrov spaces and preorders in general, providing the preorder characterizations as well as the interior and closure algebraic characterizations.
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. Arenas.
Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990s began investigating the Alexandrov topology corresponding to causal sets which arise from a preorder defined on spacetime modeling causality.
Read more about this topic: Alexandrov Topology
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