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Essentially unique
From Wikipedia, the free encyclopedia
In mathematics, the term essentially unique is used to indicate that while some object is not the only one that satisfies certain properties, all such objects are "the same" in some sense appropriate to the circumstances. This notion of "sameness" is often formalized using an equivalence relation.
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Suppose that we seek to classify all possible groups. We would find that there is an essentially unique group containing exactly 3 elements, the cyclic group of order three. No matter how we choose to write those three elements and denote the group operation, all such groups are isomorphic, hence, "the same".
Suppose that we seek a translation-invariant, strictly positive, locally finite measure on the real line. The solution to this problem is essentially unique: any such measure must be a constant multiple of Lebesgue measure. Specifying that the measure of the unit interval should be 1 then determines the solution uniquely.
Suppose that we seek to classify all two-dimensional, orientable, compact, simply connected manifolds. We would find an essentially unique solution to this problem: the 2-sphere. In this case, the solution is unique up to homeomorphism.