The supremumPlanetmathPlanetmath of a set X having a partial orderMathworldPlanetmath is the least upper bound of X (if it exists) and is denoted supX.

Let A be a set with a partial order , and let XA. Then s=supX if and only if:

  1. 1.

    For all xX, we have xs (i.e. s is an upper bound).

  2. 2.

    If s meets condition 1, then ss (s is the least upper bound).

There is another useful definition which works if A= with the usual order on , supposing that s is an upper bound:

s=supX if and only if ε>0,xX:s-ε<x.

Note that it is not necessarily the case that supXX. Suppose X=]0,1[, then supX=1, but 1X.

Note also that a set may not have an upper bound at all.

Title supremum
Canonical name Supremum
Date of creation 2013-03-22 11:48:12
Last modified on 2013-03-22 11:48:12
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 06A06
Related topic Infimum
Related topic MinimalAndMaximalNumber
Related topic InfimumAndSupremumForRealNumbers
Related topic ExistenceOfSquareRootsOfNonNegativeRealNumbers
Related topic LinearContinuum
Related topic NondecreasingSequenceWithUpperBound
Related topic EssentialSupremum