Many physical theories start with the assumption that a physical system can be described by a 1form of Action, A, such as A= pdq  Hdt. Following Cartan's methods other pforms can be constructed by forming the exterior derivative, dA, and all possible exterior products, A, dA, A^dA, dA^dA, ...
The largest nonzero element of this Pfaff sequence determines the class, or Pfaff dimension, of the given exterior form. Remarkably, for 4 dimensions the set of elements, A, AUdA, A^dA,dAUdA,... forms a topological basis. The truthtable for this topology, defined as Cartan's topology, is given in the details
If the topological torsion vanishes, and the Pfaff dimension is less than three
(A^dA=0), it turns out that the Cartan Topology is connected. However, if the
topological torsion does not vanish, then the Cartan topology is a disconnected
topology.
Recall that a topology is a method of determining a topological structure, whose
utility is in giving a definition to what is meant by continuous. As it
is impossible to construct a continuous map from a connected to a disconnected
topology, it follows that a transition from a laminar flow (A^dA=0) to a
turbulent flow (A^dA>0) must be a discontinuous process. On the other hand, a
continous map from a disconnected topology to a connected topology is possible,
therefor the decay of turbulence can be studied in terms of continuous
processes.
