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By "isomorphism" I mean any structure-preserving map with a structure-preserving inverse.

(Please accept my advance apology if this question is out of bounds. I sense that it's borderline, but I'm hoping it'll be considered in better taste than the typical "What is your favorite X?" question. I think a collection of great isomorphisms would be interesting because of what isomorphisms uniquely have the power to do: reveal deep and astonishing connections between seemingly unrelated fields of study; open a channel through which techniques and ideas can pass between disciplines; collapse two or many problems into one.)

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    "By 'isomorphism' I mean any bijective, structure-preserving map." You shouldn't. Under that definition, the topological space $[0,1)$ is isomorphic to the unit circle. An isomorphism is a structure-preserving map with a structure-preserving inverse.2010-08-24
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    Right you are. Fixed.2010-08-24

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