Cantor's intersection theorem: Difference between revisions

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{{short description|On decreasing nested sequences of non-empty compact sets}}

'''Cantor's intersection theorem''',<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Cantor's Intersection Theorem |url=https://mathworld.wolfram.com/CantorsIntersectionTheorem.html |access-date=2025-06-20 |website=mathworld.wolfram.com |language=en}}</ref> also called '''Cantor's nested intervals theorem''',<ref>{{Cite book |last=Segura |first=Julio |url=https://www.google.com.br/books/edition/An_Eponymous_Dictionary_of_Economics/Z6Oy4L-6LSwC |title=An Eponymous Dictionary of Economics: A Guide to Laws and Theorems Named After Economists |last2=Braun |first2=Carlos Rodríguez |date=2004-01-01 |publisher=Edward Elgar Publishing |isbn=978-1-84542-360-5 |pages=38 |language=en}}</ref><ref>{{Cite book |last=Denlinger |first=Charles G. |url=https://www.google.com.br/books/edition/Elements_of_Real_Analysis/CeTkVSXlj4cC |title=Elements of Real Analysis |date=2010-05-08 |publisher=Jones & Bartlett Publishers |isbn=978-1-4496-5993-6 |pages=103 |language=en}}</ref> refers to two closely related theorems in [[general topology]] and [[real analysis]], named after [[Georg Cantor]], about intersections of decreasing nested [[sequence]]s of non-empty compact sets.

'''Cantor's intersection theorem''' refers to two closely related theorems in [[general topology]] and [[real analysis]], named after [[Georg Cantor]], about intersections of decreasing nested [[sequence|sequences]] of non-empty compact sets.

==Topological statement==

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This version follows from the general topological statement in light of the [[Heine&ndash;Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. However, it is typically used as a lemma in proving said theorem, and therefore warrants a separate proof.

As an example, if <math>C_k=[0,1/k]</math>, the intersection over <math>(C_k)_{k \geq 0}</math> is&nbsp;<math>\{0\}</math>. On the other hand, both the sequence of open bounded sets <math>C_k=(0,1/k)</math> and the sequence of unbounded closed sets <math>C_k=[k,\infty)</math> have empty intersection. All these sequences are properly nested.

This version of the theorem generalizes to <math>\mathbf{R}^n</math>, the set of <math>n</math>-element vectors of real numbers, but does not generalize to arbitrary [[metric space]]s. For example, in the space of [[rational number]]s, the sets

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''Proof.'' Each nonempty, closed, and bounded subset <math>C_k\subset\mathbb{R}</math> admits a minimal element <math>x_k</math>. Since for each <math>k</math>, we have

:<math>x_{k+1} \in C_{k+1} \subset C_k</math>,

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== References ==

{{Reflist}}

* {{MathWorld | urlname=CantorsIntersectionTheorem | title=Cantor's Intersection Theorem}}

* Jonathan Lewin. An interactive introduction to mathematical analysis. Cambridge University Press. {{ISBN|0-521-01718-1}}. Section 7.8.