Content deleted Content added
Rich Smith (talk | contribs) CheckWiki Fixes (and other AWB fixes) |
|||
(12 intermediate revisions by 11 users not shown) | |||
Line 1:
{{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.
==Topological
'''Theorem.''' ''Let <math> S </math> be a [[
:<math>C_0 \supset C_1 \supset \cdots \supset C_n \supset C_{n+1} \supset \cdots, </math>
Line 17:
Since <math>C_0\subset S</math> is compact and <math>\{U_k \vert k \geq 0\}</math> is an open cover (on <math>C_0</math>) of <math>C_0</math>, a finite cover <math>\{U_{k_1}, U_{k_2}, \ldots, U_{k_m}\}</math> can be extracted. Let <math>M=\max_{1\leq i\leq m} {k_i}</math>. Then <math>{\textstyle \bigcup_{i = 1}^m U_{k_i}}=U_M</math> because <math>U_1\subset U_2\subset\cdots\subset U_n\subset U_{n+1}\cdots</math>, by the nesting hypothesis for the collection <math>(C_k)_{k \geq 0}</math>. Consequently, <math>C_0={\textstyle \bigcup_{i = 1}^m U_{k_i}} = U_M</math>. But then <math>C_M=C_0\setminus U_M=\emptyset</math>, a contradiction. [[Q.E.D.|∎]]
==Statement for
The theorem in real analysis draws the same conclusion for [[closed set|closed]] and [[bounded set|bounded]] subsets of the set of [[real number]]s <math>\mathbb{R}</math>. It states that a decreasing nested sequence <math>(C_k)_{k \geq 0}</math> of non-empty, closed and bounded subsets of <math>\mathbb{R}</math> has a non-empty intersection.
This version follows from the general topological statement in light of the [[Heine–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 <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
Line 43:
:
''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>,
Line 49:
:<math>x_k \le x_{k+1}</math>,
so <math>(x_k)_{k \geq 0}</math> is an increasing sequence contained in the bounded set <math>C_0</math>. The [[monotone convergence theorem]] for bounded sequences of real numbers now guarantees the existence of a [[Limit of a sequence|limit point]]
:<math>x=\lim_{k\to \infty} x_k.</math>
For fixed <math>k</math>, <math>x_j\in C_k</math> for all <math>j\geq k</math>, and since <math>C_k</math> is closed and <math>x</math> is a
== Variant in complete metric spaces ==
Line 75:
A converse to this theorem is also true: if <math>X</math> is a metric space with the property that the intersection of any nested family of non-empty closed subsets whose diameters tend to zero is non-empty, then <math>X</math> is a complete metric space. (To prove this, let <math>(x_k)_{k \geq 1}</math> be a Cauchy sequence in <math>X</math>, and let <math>C_k</math> be the closure of the tail <math>(x_j)_{j \geq k}</math> of this sequence.)
== See also ==
* [[Kuratowski's intersection theorem]]
* [[Helly's theorem]] - another theorem on intersection of sets.
== 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.
|