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