In mathematics, an upper set is a subset Y of a given set X such that, for all elements x and y, if x is less than or equal to y and x is an element of Y, then y is also in Y. More formally,
An upper set is sometimes denoted with the 'up arrow', thus:
A related notion is Lower set.
References
Blanck, J. (2000) "Domain representations of topological spaces". Theoretical Computer Science, 247, 229–255.
In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S.
The difference between the supremum of a set and the greatest element of a set may not be immediately obvious.
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-upper-bound of the set.
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S.
On the other hand, a set may have many different upper and lower bounds, and hence one is usually interested in picking out specific elements from the sets of upper or lower bounds.
A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds.
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