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- Least-upper-bound_property abstract "In mathematics, the least-upper-bound property (sometimes the completeness or supremum property) is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound (or supremum).The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers (see least upper bound axiom), and it is also intimately related to the construction of the real numbers using Dedekind cuts.In order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum.".
- Least-upper-bound_property thumbnail Illustration_of_supremum.svg?width=300.
- Least-upper-bound_property wikiPageID "22038671".
- Least-upper-bound_property wikiPageLength "10554".
- Least-upper-bound_property wikiPageOutDegree "56".
- Least-upper-bound_property wikiPageRevisionID "682158065".
- Least-upper-bound_property wikiPageWikiLink Bolzano–Weierstrass_theorem.
- Least-upper-bound_property wikiPageWikiLink Category:Articles_containing_proofs.
- Least-upper-bound_property wikiPageWikiLink Category:Order_theory.
- Least-upper-bound_property wikiPageWikiLink Category:Real_analysis.
- Least-upper-bound_property wikiPageWikiLink Cauchy_sequence.
- Least-upper-bound_property wikiPageWikiLink Completeness_(order_theory).
- Least-upper-bound_property wikiPageWikiLink Completeness_axiom.
- Least-upper-bound_property wikiPageWikiLink Completeness_of_the_real_numbers.
- Least-upper-bound_property wikiPageWikiLink Construction_of_the_real_numbers.
- Least-upper-bound_property wikiPageWikiLink Continuous_function.
- Least-upper-bound_property wikiPageWikiLink Cover_(topology).
- Least-upper-bound_property wikiPageWikiLink Dedekind_cut.
- Least-upper-bound_property wikiPageWikiLink Dense_order.
- Least-upper-bound_property wikiPageWikiLink Empty_set.
- Least-upper-bound_property wikiPageWikiLink Extreme_value_theorem.
- Least-upper-bound_property wikiPageWikiLink Heine–Borel_theorem.
- Least-upper-bound_property wikiPageWikiLink Infimum_and_supremum.
- Least-upper-bound_property wikiPageWikiLink Intermediate_value_theorem.
- Least-upper-bound_property wikiPageWikiLink Irrational_number.
- Least-upper-bound_property wikiPageWikiLink Least_upper_bound.
- Least-upper-bound_property wikiPageWikiLink Least_upper_bound_axiom.
- Least-upper-bound_property wikiPageWikiLink Limit_point.
- Least-upper-bound_property wikiPageWikiLink Linear_continuum.
- Least-upper-bound_property wikiPageWikiLink Linearly_ordered_set.
- Least-upper-bound_property wikiPageWikiLink List_of_real_analysis_topics.
- Least-upper-bound_property wikiPageWikiLink Mathematics.
- Least-upper-bound_property wikiPageWikiLink Nested_intervals.
- Least-upper-bound_property wikiPageWikiLink Nested_intervals_theorem.
- Least-upper-bound_property wikiPageWikiLink Nonempty.
- Least-upper-bound_property wikiPageWikiLink Open_set.
- Least-upper-bound_property wikiPageWikiLink Order_theory.
- Least-upper-bound_property wikiPageWikiLink Partially_ordered_set.
- Least-upper-bound_property wikiPageWikiLink Rational_number.
- Least-upper-bound_property wikiPageWikiLink Real_analysis.
- Least-upper-bound_property wikiPageWikiLink Real_number.
- Least-upper-bound_property wikiPageWikiLink Root_of_a_function.
- Least-upper-bound_property wikiPageWikiLink Sequence.
- Least-upper-bound_property wikiPageWikiLink Set_(mathematics).
- Least-upper-bound_property wikiPageWikiLink Subsequence.
- Least-upper-bound_property wikiPageWikiLink Subset.
- Least-upper-bound_property wikiPageWikiLink Supremum.
- Least-upper-bound_property wikiPageWikiLink Theorem.
- Least-upper-bound_property wikiPageWikiLink Total_order.
- Least-upper-bound_property wikiPageWikiLink Undergraduate_Texts_in_Mathematics.
- Least-upper-bound_property wikiPageWikiLink Upper_and_lower_bounds.
- Least-upper-bound_property wikiPageWikiLink Upper_bound.
- Least-upper-bound_property wikiPageWikiLink Zero_of_a_function.
- Least-upper-bound_property wikiPageWikiLink File:Illustration_of_supremum.svg.
- Least-upper-bound_property wikiPageWikiLinkText "Least-upper-bound property".
- Least-upper-bound_property wikiPageWikiLinkText "least upper bound property".
- Least-upper-bound_property wikiPageWikiLinkText "least-upper-bound property".
- Least-upper-bound_property hasPhotoCollection Least-upper-bound_property.
- Least-upper-bound_property wikiPageUsesTemplate Template:Cite_book.
- Least-upper-bound_property wikiPageUsesTemplate Template:Main.
- Least-upper-bound_property wikiPageUsesTemplate Template:Math.
- Least-upper-bound_property wikiPageUsesTemplate Template:Reflist.
- Least-upper-bound_property subject Category:Articles_containing_proofs.
- Least-upper-bound_property subject Category:Order_theory.
- Least-upper-bound_property subject Category:Real_analysis.
- Least-upper-bound_property hypernym Property.
- Least-upper-bound_property type Building.
- Least-upper-bound_property type Field.
- Least-upper-bound_property type Proof.
- Least-upper-bound_property comment "In mathematics, the least-upper-bound property (sometimes the completeness or supremum property) is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound (or supremum).The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness.".
- Least-upper-bound_property label "Least-upper-bound property".
- Least-upper-bound_property sameAs Supremumseigenschaft.
- Least-upper-bound_property sameAs m.0ds37rv.
- Least-upper-bound_property sameAs Q1254734.
- Least-upper-bound_property sameAs Q1254734.
- Least-upper-bound_property wasDerivedFrom Least-upper-bound_property?oldid=682158065.
- Least-upper-bound_property depiction Illustration_of_supremum.svg.
- Least-upper-bound_property isPrimaryTopicOf Least-upper-bound_property.