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supremum (Definition)

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 $\leq$ and let $X \subseteq A$ Then $s = \sup X$ if and only if:

1.
For all $x\in X$ we have $x \leq s$ (i.e. $s$ is an upper bound).
2.
If $s^{\prime}$ meets condition 1, then $s \leq s^{\prime}$ ($s$ is the least upper bound).

There is another useful definition which works if $A = \mathbb{R}$ with $\leq$ 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.




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"supremum" is owned by Cosmin. [ full author list (2) | owner history (1) ]
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See Also: infimum, minimal and maximal number, infimum and supremum for real numbers, existence of square roots of non-negative real numbers, linear continuum, limit of nondecreasing sequence, essential supremum

Keywords:  real analysis

Attachments:
arbitrary join (Definition) by CWoo
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Cross-references: upper bound, least upper bound, partial order
There are 41 references to this entry.

This is version 6 of supremum, born on 2001-10-18, modified 2007-08-11.
Object id is 340, canonical name is Supremum.
Accessed 23089 times total.

Classification:
AMS MSC06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

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