Cantor's intersection theorem: Difference between revisions

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

In [[real analysis]], a bran, '''Cantor's intersection theorem''', named after [[Georg Cantor]], is a theorem related to [[compact set]]s in '''R''', the set of [[real number]]s. It states that a decreasing nested [[sequence]] of non-empty, [[closed set|closed]] and [[bounded set|bounded]] subsets of '''R''' has nonempty intersection. In other words, supposing {''C''_''k''} is a sequence of non-empty, closed and bounded sets satisfying

'''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 statement==

:<math>C_0 \supseteq C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots, \, </math>

'''Theorem.''' ''Let <math> S </math> be a [[topological space]]. A decreasing nested sequence of non-empty compact, closed subsets of <math>S</math> has a non-empty intersection. In other words, supposing <math>(C_k)_{k \geq 0}</math> is a sequence of non-empty compact, closed subsets of S satisfying''

:<math>C_0 \supset C_1 \supset \cdots \supset C_n \supset C_{n+1} \supset \cdots, </math>

it follows that

''it follows that''

:<math>\left(\bigcap_{k} C_k\right) \neq \emptyset. \, </math>

:<math>\left(\bigcap_{k = 0}^\infty C_k\right) \neq \emptyset. \, </math>

The result is typically used as a lemma in proving the [[Heine&ndash;Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine&ndash;Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of '''R''' has nonempty intersection.

The closedness condition may be omitted in situations where every compact subset of <math>S</math> is closed, for example when <math>S</math> is [[Hausdorff space|Hausdorff]].

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

'''Proof.''' Assume, by way of contradiction, that <math>{\textstyle \bigcap_{k = 0}^\infty C_k}=\emptyset</math>. For each <math>k</math>, let <math>U_k=C_0\setminus C_k</math>. Since <math>{\textstyle \bigcup_{k = 0}^\infty U_k}=C_0\setminus {\textstyle \bigcap_{k = 0}^\infty C_k}</math> and <math>{\textstyle \bigcap_{k = 0}^\infty C_k}=\emptyset</math>, we have <math>{\textstyle \bigcup_{k = 0}^\infty U_k}=C_0</math>. Since the <math>C_k</math> are closed relative to <math>S</math> and therefore, also closed relative to <math>C_0</math>, the <math>U_k</math>, their set complements in <math>C_0</math>, are open relative to <math>C_0</math>.

The theorem generalizes to '''R'''^''n'', the set of ''n''-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

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 real numbers==

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.

TheThis resultversion isfollows typicallyfrom usedthe asgeneral atopological lemmastatement in provinglight of the [[Heine&ndash;Borel theorem]], which states that sets of real numbers are compact if and only if they are closed and bounded. ConverselyHowever, if the Heine&ndash;Borel theoremit is known,typically then it can be restatedused as: a decreasinglemma nestedin sequenceproving ofsaid non-emptytheorem, compactand subsetstherefore ofwarrants '''R''' hasa nonemptyseparate intersectionproof.

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

TheThis version of the theorem generalizes to '''R'''<supmath>''\mathbf{R}^n''</supmath>, 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

: <math>C_k = [\sqrt{2}, \sqrt{2}+1/k] = (\sqrt{2}, \sqrt{2}+1/k)</math>

are closed and bounded, but their intersection is empty.

Note that this contradicts neither the topological statement, as the sets <math>C_k</math> are not compact, nor the variant below, as the rational numbers are not complete with respect to the usual metric.

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.

'''Theorem.''' ''Let'' <math>(C_k)_{k \geq 0}</math> ''be a sequence of non-empty, closed, and bounded subsets of'' <math>\mathbb{R}</math> ''satisfying''

== Proof ==

Consider the sequence (''a''_''k'') where ''a''_''k'' is the [[infimum]] over the non-empty ''C''_''k''. Because ''C''_''k'' is closed, ''a''_''k'' belongs to ''C''_''k''; because the sets are decreasing nested, the sequence is monotonic increasing. Because it is also bounded (being contained in the bounded set ''C''₁), it must converge to some limit&nbsp;''L''. Choose any ''j''&nbsp;≥&nbsp;1; the subsequence of (''a''_''k'') for ''k''&nbsp;≥&nbsp;''j'' is contained in C_''j'' and converges to&nbsp;''L''. Since ''C''_''j'' is closed, L lies in&nbsp;''C''_''j''. Since this is true for all ''j'', ''L'' lies in all ''C''_''j'', and so in their intersection.

:<math>C_0 \supseteqsupset C_1 \supseteqsupset \cdots C_kC_n \supseteqsupset C_{kn+1} \cdots, \,. </math>

''Then,''

:<math>\bigcap_{k = 0}^\infty C_k \neq \emptyset. </math>

:

''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>,

it follows that

:<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 limit point, it follows that <math>x\in C_k</math>. Our choice of <math>k</math> is arbitrary, hence <math>x</math> belongs to <math>{\textstyle \bigcap_{k = 0}^\infty C_k}</math> and the proof is complete. ∎

== Variant in complete metric spaces ==

In a [[complete metric space]], the following variant of Cantor's intersection theorem holds.

'''Theorem.''' ''Suppose that <math>X</math> is a complete metric space, and <math>(C_k)_{k \geq 1}</math> is a sequence of non-empty closed nested subsets of <math>X</math> whose [[diameter]]s tend to zero:''

:<math>\lim_{k\to\infty} \operatorname{diam}(C_k) = 0,</math>

''where <math>\operatorname{diam}(C_k)</math> is defined by''

:<math>\operatorname{diam}(C_k) = \sup\{d(x,y) \mid x,y\in C_k\}.</math>

''Then the intersection of the <math>C_k</math> contains exactly one point:''

:<math>\bigcap_{k=1}^\infty C_k = \{x\}</math>

''for some <math>x \in X</math>.''

''Proof (sketch).'' Since the diameters tend to zero, the diameter of the intersection of the <math>C_k</math> is zero, so it is either empty or consists of a single point. So it is sufficient to show that it is not empty. Pick an element <math>x_k\in C_k</math> for each <math>k</math>. Since the diameter of <math>C_k</math> tends to zero and the <math>C_k</math> are nested, the <math>x_k</math> form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point <math>x</math>. Since each <math>C_k</math> is closed, and <math>x</math> is a limit of a sequence in <math>C_k</math>, <math>x</math> must lie in <math>C_k</math>. This is true for every <math>k</math>, and therefore the intersection of the <math>C_k</math> must contain <math>x</math>. ∎

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.)

== ProofSee 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.

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