Beyond Topologies, Part I

TL;DR

This paper discusses the limitations of traditional HKB topologies and explores extended topological concepts, especially in relation to topological type processes in measure, integration, and PDEs.

Contribution

It introduces a generalized topology framework that encompasses previous extensions, connecting it to solutions of nonlinear PDEs beyond classical HKB topologies.

Findings

Classical Moore-Smith conditions cannot be satisfied within HKB topologies.

Extended topologies can incorporate topological type processes in measure and PDE theories.

The proposed generalized topology unifies previous approaches and broadens applicability.

Abstract

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 {\it 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 {\it four Moore-Smith conditions}, can {\it 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 {\it 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.