supremum

The supremum of a set $X$ having a partial order is the least upper bound of $X$ (if it exists) and is denoted $\sup{X}$ .

Let $A$ be a set with a partial order $\leqslant$ , and let $X\subseteq A$ . Then $s=\sup X$ if and only if:

1.

For all $x\in X$ , we have $x\leqslant s$ (i.e. $s$ is an upper bound).
2.

If $s^{\prime}$ meets condition 1, then $s\leqslant s^{\prime}$ ( $s$ is the least upper bound).

There is another useful definition which works if $A=\mathbb{R}$ with $\leqslant$ the usual order on $\mathbb{R}$ , supposing that s is an upper bound:

s=\sup X\text{ if and only if }\forall\varepsilon>0,\exists x\in X:s-% \varepsilon<x.

Note that it is not necessarily the case that $\sup X\in X$ . Suppose $X={]0,1[}$ , then $\sup X=1$ , but $1\not\in X$ .

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