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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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closed as not constructive by Pete L. Clark, Aryabhata, Rasmus, Tom Boardman, Qiaochu Yuan Aug 24 '10 at 12:01

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

"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. – Pete L. Clark Aug 24 '10 at 5:02
Right you are. Fixed. – Jason Orendorff Aug 24 '10 at 6:10

The natural isomorphism between a finite-dimensional space and its double-dual.

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Isomorphism between the infinite spiral given by Riemann surface of $\log(z)$ and the punctured sphere.

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To follow previous answers, I would say the isomorphism between a finite-dimensional vector space and its dual, even though it isn't natural (MacLane & Birkhoff 1999, §VI.4, from Wikipedia).

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I like the idea that an $n$ dimensional vector space over the field $\mathbb{C}$ is isomorphic to $\mathbb{C}^n$

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What's the isomorphism between any $n$-dimensional vector space to $\mathbb C^n$? – Rasmus Aug 24 '10 at 8:45
Sorry poor choice of words on my part. – Digital Gal Aug 24 '10 at 15:16

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