The infimumMathworldPlanetmathPlanetmath of a set S is the greatest lower bound of S and is denoted inf(S).

Let A be a set with a partial orderMathworldPlanetmath , and let SA. For any xA, x is a lower bound of S if xy for any yS. The infimum of S, denoted inf(S), is the greatest such lower bound; that is, if b is a lower bound of S, then binf(S).

Note that it is not necessarily the case that inf(S)S. Suppose S=(0,1); then inf(S)=0, but 0S.

Also note that a set does not necessarily have an infimum. See the attachments to this entry for examples.

Title infimum
Canonical name Infimum
Date of creation 2013-03-22 11:48:09
Last modified on 2013-03-22 11:48:09
Owner vampyr (22)
Last modified by vampyr (22)
Numerical id 11
Author vampyr (22)
Entry type Definition
Classification msc 06A06
Classification msc 03D20
Related topic SupremumMathworldPlanetmath
Related topic LebesgueOuterMeasure
Related topic MinimalAndMaximalNumber
Related topic InfimumAndSupremumForRealNumbers
Related topic NondecreasingSequenceWithUpperBound