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| 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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| A related notion is a [[universal property]], where an object is not only essentially unique, but unique ''up to a unique [[isomorphism]]'' (meaning that it has trivial [[automorphism group]]). In general given two isomorphic examples of an essentially unique object, there is no ''natural'' (unique) isomorphism between them.
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| ==Examples==
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| ===Set theory===
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| Most basically, there is an essentially unique set of any given [[cardinality]], whether one labels the elements <math>\{1,2,3\}</math> or <math>\{a,b,c\}</math>.
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| In this case the non-uniqueness of the isomorphism (does one match 1 to ''a'' or to ''c''?) is reflected in the [[symmetric group]].
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| On the other hand, there is an essentially unique ''ordered'' set of any given finite cardinality: if one writes <math>\{1 < 2 < 3\}</math> and <math>\{a< b< c\}</math>, then the only order-preserving isomorphism maps 1 to ''a,'' 2 to ''b,'' and 3 to ''c.''
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| ===Group theory===
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| Suppose that we seek to classify all possible [[group (mathematics)|group]]s. 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".
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| On the other hand, there is not an essentially unique group with exactly 4 elements, as there are two non-isomorphic examples: the cyclic group of order 4 and the [[Klein four group]].
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| ===Measure theory===
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| Suppose that we seek a [[translation (mathematics)|translation]]-[[invariant measure|invariant]], [[strictly positive measure|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.
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| ===Topology===
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| Suppose that we seek to classify all two-dimensional, [[compact space|compact]], [[simply connected]] [[manifold]]s. We would find an essentially unique solution to this problem: the [[2-sphere]]. In this case, the solution is unique up to [[homeomorphism]].
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| ===Lie theory===
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| A [[maximal compact subgroup]] of a [[semisimple Lie group]] may not be unique, but is unique up to conjugation.
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| ==See also==
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| * [[Classification theorem]]
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| * [[Universal property]]
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| {{DEFAULTSORT:Essentially Unique}}
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| [[Category:Mathematical terminology]]
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