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Consequently, the supremum is also referred to as the least upper bound (or LUB). [1] The infimum is, in a precise sense, dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration.
Roughly speaking, these functions map the supremum/infimum of a set to the supremum/infimum of the image of the set. Depending on the type of sets for which a function satisfies this property, it may preserve finite, directed, non-empty, or just arbitrary suprema or infima.
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.
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Infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a maximal element. However, if an infimum or supremum does exist, it is unique. Is the below not a counterexample. (if a rather odd one)
Then f preserves the supremum of S if the set f(S) = {f(x) | x in S} has a least upper bound in Q which is equal to f(s), i.e. f(sup S) = sup f(S) This definition consists of two requirements: the supremum of the set f(S) exists and it is equal to f(s). This corresponds to the abovementioned parallel to category theory, but is not always ...
The supremum of the ordinals that are the image of a function from ω ω (usually in models where the axiom of choice is not assumed) κ 1. Often used for a cardinal, especially the critical point of an elementary embedding 2. The Erdős cardinal κ(α) is the smallest cardinal such that κ(α) → (α) < ω λ 1. Often used for a cardinal 2.