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- Cantor's_intersection_theorem abstract "In real analysis, a branch of mathematics, Cantor's intersection theorem, named after Georg Cantor, is a theorem related to compact sets of a compact space . It states that a decreasing nested sequence of non-empty compact subsets of has nonempty intersection. In other words, supposing {Ck} is a sequence of non-empty, closed and bounded sets satisfyingit follows thatThe result is typically used as a lemma in proving the Heine–Borel theorem, which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine–Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of a compact space has nonempty intersection.As an example, if Ck = [0, 1/k], the intersection over {Ck} is {0}. On the other hand, both the sequence of open bounded sets Ck = (0, 1/k) and the sequence of unbounded closed sets Ck = [k, ∞) have empty intersection. All these sequences are properly nested.The theorem generalizes to Rn, the set of n-element vectors of real numbers, but does not generalize to arbitrary metric spaces. For example, in the space of rational numbers, the sets are closed and bounded, but their intersection is empty.A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. In fact, the Cantor set contains uncountably many points.".
- Cantor's_intersection_theorem wikiPageID "28046776".
- Cantor's_intersection_theorem wikiPageRevisionID "606548278".
- Cantor's_intersection_theorem hasPhotoCollection Cantor's_intersection_theorem.
- Cantor's_intersection_theorem title "Cantor's Intersection Theorem".
- Cantor's_intersection_theorem urlname "CantorsIntersectionTheorem".
- Cantor's_intersection_theorem subject Category:Articles_containing_proofs.
- Cantor's_intersection_theorem subject Category:Compactness_theorems.
- Cantor's_intersection_theorem subject Category:Real_analysis.
- Cantor's_intersection_theorem subject Category:Theorems_in_calculus.
- Cantor's_intersection_theorem type Abstraction100002137.
- Cantor's_intersection_theorem type Communication100033020.
- Cantor's_intersection_theorem type CompactnessTheorems.
- Cantor's_intersection_theorem type Message106598915.
- Cantor's_intersection_theorem type Proposition106750804.
- Cantor's_intersection_theorem type Statement106722453.
- Cantor's_intersection_theorem type Theorem106752293.
- Cantor's_intersection_theorem type TheoremsInCalculus.
- Cantor's_intersection_theorem comment "In real analysis, a branch of mathematics, Cantor's intersection theorem, named after Georg Cantor, is a theorem related to compact sets of a compact space . It states that a decreasing nested sequence of non-empty compact subsets of has nonempty intersection.".
- Cantor's_intersection_theorem label "Cantor's intersection theorem".
- Cantor's_intersection_theorem label "Principio di localizzazione di Cantor".
- Cantor's_intersection_theorem sameAs Principio_di_localizzazione_di_Cantor.
- Cantor's_intersection_theorem sameAs 칸토어의_교점_정리.
- Cantor's_intersection_theorem sameAs m.0cm8l2y.
- Cantor's_intersection_theorem sameAs Q1050203.
- Cantor's_intersection_theorem sameAs Q1050203.
- Cantor's_intersection_theorem sameAs Cantor's_intersection_theorem.
- Cantor's_intersection_theorem wasDerivedFrom Cantor's_intersection_theorem?oldid=606548278.
- Cantor's_intersection_theorem isPrimaryTopicOf Cantor's_intersection_theorem.