One primitive topological property is the concept of the number of parts, or components, of a set. If the number of parts changes during an evolutionary process, then the topology of the initial state is not the same as the topology of the final state. The number of parts is not the only topological property that can change during an evolutionary process. A simple intuitive description of topological evolution can be expressed in terms of the operation of cutting into separable parts, and pasting of parts together. Although both processes effect topological change, the cutting process is a discontinuous process, where the pasting process is continuous. As demonstrated herein, Cartan's exterior calculus permits analytic progress to be made in the understanding of irreversible processes that involve continous topological change. As of this date (with notable exceptions - see the discussion on defects and singularities) the discontinuous processes have been little studied in terms of Cartan's techniques.

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