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Are there any examples other than using dimension for vector spaces where the easiest way to show that two objects are isomorphic is by using a classification theorem and showing that they must both be in the same class? (homeomorphisms count too)

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Finite sets, of course. – Qiaochu Yuan Aug 3 '10 at 3:33
Finite fields, algebraically closed fields, Hilbert spaces. – Kevin Ventullo Aug 3 '10 at 5:02
I have a hard time understanding this question. Metamathematically speaking, isn't that a standard situation? Typically, a classification scheme (if available) involves constructing a robust set of invariants and isomorphism of two general objects is checked by computing and comparing them. – Victor Protsak Aug 3 '10 at 5:05
That's not the situation for isomorphism of groups given by presentations. – Ryan Budney Aug 3 '10 at 18:11
up vote 3 down vote accepted

Genus for surfaces would be a simple example.

Connectedness for compact $1$-dimensional manifolds would be another!

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