# 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 $\leq$, and let $S\subseteq A$. For any $x\in A$, $x$ is a lower bound of $S$ if $x\leq y$ 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 $b\leq\inf(S)$.

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

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