A simple definition of topology is that it is the study of properties that do NOT depend upon size or shape. The most fundamentalof topological properties is the number -- the number of parts, the number of intersections, the number of links, the number of holes, the number of dimensions..... Topological evolution is the study of when and how these numbers change.
Similarly, geometry may be said to be the study of properties that depend upon
size and shape. It is extraordinary, but most current scientific and
engineering concepts are based upon the geometric tradition, where the concept
of number is constant, and the notion of topological evolution is ignored.
Non-uniqueness and discontinuities are abhored. Yet there is evidence that
irreversibility and aging imply changing topology.
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