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单词 Supremum
释义

supremum


The supremumPlanetmathPlanetmath of a set X having a partial orderMathworldPlanetmath is the least upper bound of X (if it exists) and is denoted supX.

Let A be a set with a partial order , and let XA. Then s=supX if and only if:

  1. 1.

    For all xX, we have xs (i.e. s is an upper bound).

  2. 2.

    If s meets condition 1, then ss (s is the least upper bound).

There is another useful definition which works if A= with the usual order on , supposing that s is an upper bound:

s=supX if and only if ε>0,xX:s-ε<x.

Note that it is not necessarily the case that supXX. Suppose X=]0,1[, then supX=1, but 1X.

Note also that a set may not have an upper bound at all.

Titlesupremum
Canonical nameSupremum
Date of creation2013-03-22 11:48:12
Last modified on2013-03-22 11:48:12
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id11
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 06A06
Related topicInfimum
Related topicMinimalAndMaximalNumber
Related topicInfimumAndSupremumForRealNumbers
Related topicExistenceOfSquareRootsOfNonNegativeRealNumbers
Related topicLinearContinuum
Related topicNondecreasingSequenceWithUpperBound
Related topicEssentialSupremum
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更新时间:2025/5/4 17:40:44