Cartan's topological structure

A system of di®erential forms will establish a topology and a topological structure on a domain of independent variables such that is possible to determine which maps or processes acting on the system are continuous. Perhaps the most simple topology is that generated by the existence of a single 1-form of Action, its Pfa® sequence of exterior di®erentials, and their intersections. In such a topology the exterior derivative becomes a limit point generator in the sense of Kuratowski. The utilization of such techniques in physical systems is examined. A key feature of the Cartan topology is determined by the Pfa® dimension (representing the minimum number of functions to describe the 1-form generator). In particular, when the Pfa® dimension is 3 or more the Cartan topology becomes a disconnected topology, with the existence of topological torsion and topological parity. Most classical physical applications are constrained to cases where the Pfa® dimension is 2 or less, for such is the domain of unique integrability. The more interesting domain of non-unique solutions requires the existence of topological torsion, and can lead to an understanding of irreversible processes without the use of statistics.