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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

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

4 Answers 4

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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