Topological Equations Theory

This work emerging body of work provides the theoretical foundations, the classification schema, and the solution methodologies for the topological equations. These equations are new characterisations of them in terms of such topological structures which are then combined with mathematical relation twisted by them and a full fledged analytical framework where functional analysis and differential topology as well as category theory are presented. Existence and uniqueness theorems at the fundamental level and their consequences concerning solution spaces from which algorithmically implementable solution methods can be derived are derived. Examples from modern problems and computational challenges in mathematical physics and computational topology in light of topological equations are also demonstrated in this dissertation, where a variety of dynamical systems, geometric reconstruction and cohomology automatic are used. It extends the combination of topologies that had grown historically so completely apart and supplies a common structure, with numerous consequences for use in both pure and emergent computational applications.