Beyond Topologies, Part I

Original · EN

Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known topological type processes, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classical characterization given by the four Moore-Smith conditions, can no longer be incorporated within the usual HKB topologies. One of the most successful recent ways to go beyond HKB topologies is that developed in Beattie & Butzmann. It is shown in this work how that extended concept of topology is a particular case of the earlier one suggested and used by the first author in the study of generalized solutions of large classes of nonlinear partial differential equations.