Cryptomorphic topological structures: A computational, relation-algebraic approach

In this paper we present an approach to pointless topology that is based on the study of the membership or is-element-of relation within a typed or categorical version of the component-free calculus of relations. This allows us to study several other approaches to topology, including point-set topology, on an abstract and component-free level. In particular, we will show that topologies defined by open sets, closed sets, a family of neighbourhood systems, a topological kernel-mapping, a Kuratowski closure-mapping, or a topological Aumann contact relation are cryptomorphic concepts, i.e., each concept can bijectively be transformed into any other of these concepts. All transformations are specified via relation-algebraic expressions which, in case of set-theoretic relations (that is, in case of point-set topology) can immediately be executed by the specific purpose computer algebra system RelView.