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There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit most tightly and do not overlap. That characterizes the Lebesgue outer measure.
Likewise, a greatest element of a partially ordered set (poset) is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A), then m = b. Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal ...
The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but is not complete. The algebra of all measurable subsets of a measure space is a ℵ 1-complete Boolean algebra, but is not usually complete.
The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of ...
At 2:05 p.m., officers battle the crowd over metal bike racks on the Capitol's lower West terrace. Dale Huttle of Indiana hits two officers with an upside-down American flag on a pole, spearing ...
For all elements x and y of S, the greatest lower bound of the set {x, y} exists. The greatest lower bound of the set {x, y} is called the meet of x and y, denoted x ∧ y. Replacing "greatest lower bound" with "least upper bound" results in the dual concept of a join-semilattice. The least upper bound of {x, y} is called the join of x and y ...