infimum

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

Let $A$ be a set with a partial order $\le$, and let $S\subseteq A$. For any $x\in A$, $x$ is a lower bound of $S$ if $x\le y$ for any $y\in S$. The infimum of $S$, denoted $inf(S)$, is the greatest such lower bound; that is, if $b$ is a lower bound of $S$, then $b\le inf(S)$.

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

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	Supremum
Related topic	LebesgueOuterMeasure
Related topic	MinimalAndMaximalNumber
Related topic	InfimumAndSupremumForRealNumbers
Related topic	NondecreasingSequenceWithUpperBound