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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.
The supremum/superior/outer limit is a set that joins these accumulation sets together. That is, it is the union of all of the accumulation sets. When ordering by set inclusion, the supremum limit is the least upper bound on the set of accumulation points because it contains each of them. Hence, it is the supremum of the limit points.
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.
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function does at points (that is, the image of ), but rather by asking for the set of points where equals a specific value (that is, the preimage of under ).
A poset in which every element is the supremum of the directed set formed by the compact elements below it is called an algebraic poset. Such posets that are dcpos are much used in domain theory. As an important special case, an algebraic lattice is a complete lattice L where every element x of L is the supremum of the compact elements below x.
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . A conditionally complete lattice satisfies at least one of these properties for bounded subsets. For comparison, in a general lattice, only pairs of elements need to have a supremum and an infimum. Every non-empty finite ...