Set theory At the most basic level, there is an essentially unique set of any given
cardinality, whether one labels the elements \{1,2,3\} or \{a,b,c\}. In this case, the non-uniqueness of the isomorphism (e.g., match 1 to a or 1 to
c) is reflected in the
symmetric group. On the other hand, there is an essentially unique
totally ordered set of any given finite cardinality that is unique
up to unique isomorphism: if one writes \{1 and \{a, then the only
order-preserving isomorphism is the one which maps 1 to
a, 2 to
b, and 3 to
c. Number theory The
fundamental theorem of arithmetic establishes that the
factorization of any positive
integer into
prime numbers is essentially unique, i.e., unique up to the ordering of the prime
factors.
Group theory In the context of classification of
groups, there is an essentially unique group containing exactly 2 elements.
Measure theory There is an essentially unique measure that is
translation-
invariant,
strictly positive and
locally finite on the
real line. In fact, any such measure must be a constant multiple of
Lebesgue measure, specifying that the measure of the unit interval should be 1—before determining the solution uniquely.
Topology There is an essentially unique two-dimensional,
compact,
simply connected manifold: the
2-sphere. In this case, it is unique up to
homeomorphism. In the area of topology known as
knot theory, there is an analogue of the fundamental theorem of arithmetic: the decomposition of a knot into a sum of
prime knots is essentially unique.
Lie theory A
maximal compact subgroup of a
semisimple Lie group may not be unique, but is unique up to
conjugation.
Category theory An object that is the
limit or colimit over a given diagram is essentially unique, as there is a
unique isomorphism to any other limiting/colimiting object.
Coding theory Given the task of using 24-
bit words to store 12 bits of information in such a way that 4-bit errors can be detected and 3-bit errors can be corrected, the solution is essentially unique: the
extended binary Golay code. ==See also==