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If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ?

Isometry sounds more geometrical than topological. What would be a good name for "piecewise topology" ? If you take a topological space and break it up and put it back together in a random way, not necessarily in one piece, then what could you say about the topology of the resulting jumble ?

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    If you no longer require continuity of the "breaking up and put-together map" (which is what I interpret to be the meaning of "in a random way"), then by the definition of cardinality, any two topological spaces with continuum many points will admit a bijection as sets, and therefore you can break any topological space (with sufficiently many points) up and reassemble it into any other topology. (See also the theorem of Banach-Tarski http://en.wikipedia.org/wiki/Banach-Tarski_paradox )2010-12-01
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    What if it can only be broken into reasonable sized/shaped chunks. What would be the definition of reasonable ?2010-12-01
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    Well, sort of the point of topological spaces is that there's not much you can say about "size" or "shape". For that you need to start talking about geometry.2010-12-01
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    But there is the discipline of "Geometric topology". Topological considerations of geometric constructions or geometrical considerations of topological constructions.2010-12-04

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