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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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You might be interested in reading about Hilbert's Third Problem. Hilbert asked the question: Given two polytopes of the same volumes, can we cut one up into polytopal pieces and reassemble it to form the other. The answer is NO in dimensions $\geq 3$; there are many additional subtle obstacles.

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If the pieces are of given shapes, then this is essentially a jigsaw puzzle, perhaps in a higher dimension than 2. A mathematical term for problems of this type is "dissection".

If the pieces don't have a specific form and don't have to be put back together "in one piece", I don't think much can be proved about the "resulting jumble".

The Banach-Tarski paradox gives an example of what is possible, at least if the Axiom of Choice is allowed. This is a result that proves a ball (solid sphere) can be partitioned into a finite number of subsets that can be recombined to form two balls, each congruent to the original. (The subsets are not measurable. It is not a true paradox, but it is so called because it shows the Axiom of Choice has implications that defy common intuition.)

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    3D glass (or safer: transparent plastic) jigsaw cubes that have to be reassembled. Some of the pieces would have things embedded in them, so the whole assembly would like some stuff encased in a glass display.2010-12-01