Axiomatic systems of Alexandrov spaces

In order to study internel axiomatic systems and ordered features of Alexandrov spaces, with the help of some existed results in topology and locale theory, by restricting the related structures into Alexandrov setting, some equivalent descriptions are obtained. The results show that Alexandrov spaces are categorically isomorphic to Alexandrov neighborhood systems, Alexandrov closure operators, Alexandrov interior operators and Alexandrov derived operators; T0 Alexandrov spaces are isomorphic to posets and dual to complete-generated lattices. Alexandrov spaces can be completely characterized by neighborhood systems, closure operators, interior operators, derived system, the specialization order and the point-free order.