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单词 NumerableSet
释义

numerable set


Let X be a set. An enumeration on X is a surjection from the set of natural numbers to X.

A set X is called numerable if there is a bijectiveMathworldPlanetmathPlanetmath enumeration on X.

It is easy to show that and are numerable.

It is a standard fact that is not numerable. For, if we suppose that the numbers [0,1] were countableMathworldPlanetmath, we can arrange them in a list (given by the supposed bijection).

Representing them in a binary form, it is not hard to construct an element in [0,1], which is not in the list.

This contradictionMathworldPlanetmathPlanetmath implies that [0,1] is not numerable.

Remark. If the enumeration X is furthermore a computable functionMathworldPlanetmath, then we say that X is enumerable. There exists numerable sets that are not enumerable.

Titlenumerable set
Canonical nameNumerableSet
Date of creation2013-03-22 16:01:32
Last modified on2013-03-22 16:01:32
Ownerjuanman (12619)
Last modified byjuanman (12619)
Numerical id11
Authorjuanman (12619)
Entry typeDefinition
Classificationmsc 97A80
Related topicCalculus
Related topicTopicsOnCalculus
Related topicDenumerable
Related topicCountable
Definesenumeration
Definesenumerable
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