Cantor's intersection theorem: Difference between revisions

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

{{Underlinked|date=September 2013}}

'''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==

In [[real analysis]], a branch of mathematics, '''Cantor's intersection theorem''', named after [[Georg Cantor]], gives conditions under which an infinite intersection of nested, non-empty, sets is non-empty.

'''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>

'''Theorem 1''': If <math>(X, d)</math> is a non-trivial, [[complete metric space]] and <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_n\supset C_{n+1},\forall n</math> and <math>\lim_{n\rightarrow\infty} diam(C_n)=\sup\{d(x,y): x,y\in C_n\}\rightarrow 0</math>. Then, there exists an <math>x\in X</math> such that <math>\bigcap_{n=1}^\infty C_n = \{x\} </math>.<ref>"Real Analysis," H.L. Royden, P.M. Fitzpatrick, 4th edition, 2010, page 195</ref>

''it follows that''

'''Theorem 2''': If <math>X</math> is a [[compact (mathematics) | compact]] space and <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_n\supset C_{n+1},\forall n</math>, then <math>\bigcap_{n=1}^\infty C_n\neq\varnothing</math>.

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

Notice the differences and the similarities between the two theorem. In Theorem 2, the <math>C_n</math> are only assumed to be closed (and not compact, which is stronger) since given a compact space <math>X</math> and <math>Y\subset X</math> a closed subset, <math>Y</math> is necessarily compact. Also, in Theorem 1 the intersection is exactly 1 point, while in Theorem 2 it could contain many more points. Interestingly, a metric space having the Cantor Intersection property (i.e. the theorem above holds) is necessarily complete (for justification see below). An example of an application of this theorem is the existence of limit points for self-similar contracting fractals.<ref>Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces, T. Bedford, M. Keane and C. Series eds., Oxford Univ. Press 1991, page 225</ref>

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]].

Notice that each of the hypotheses above is essential. If the metric space were not complete, then one could construct a nested sequence of non-empty, compact sets converging to a "hole" in the space, i.e. <math>\mathbb{Q}</math> with the usual metric and the sequence of sets, <math>C_n = [\sqrt{2}, \sqrt{2}+1/n]</math>. If the sets are not closed, then one can construct sequences of nested sets which have empty intersection, i.e. <math>\mathbb{R}</math> with the collection, <math>C_n = (0,\frac{1}{n}) </math>. The collections <math>C_n = [n, \infty)</math> and <math>C_n = [-\frac{1}{n}, 1+\frac{1}{n}]</math> illustrate what may happen when the diameters do not tend to zero: the intersection may be empty, as in the first, or may contain more than a single point, as in the second.

'''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>.

== Proof ==

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.|∎]]

'''Theorem 1''':

Suppose <math>(X,d)</math> is a non-trivial, complete metric space and <math>\{C_n\}</math> is an infinite family of non-empty closed sets in <math>X</math> such that <math>C_n\supset C_{n+1},\forall n</math> and <math>\lim_{n\to\infty} diam(C_n)\rightarrow 0</math>. Naturally we would like to use the completeness so we will construct a Cauchy sequence. Since each of the <math>C_n</math> is closed, there exists a <math>y_n</math> in the interior (i.e. positive distance to anything outside <math>C_n</math>) of <math>C_n</math>. These <math>y_n</math> form a sequence. Since <math>\lim_{n\to\infty} diam(C_n)\rightarrow 0</math>, then given any positive real value, <math>\epsilon>0</math>, there exists a large <math>N</math> such that whenever <math>n\geq N</math>, <math>diam(C_n)<\epsilon</math>. Since, <math>C_n\supset C_{n+1},\forall n</math>, then given any <math>n,m\geq N</math>, <math>y_n,y_m \in C_n</math> and therefore, <math>d(y_n,y_m)<\epsilon</math>. Thus, the <math>y_n</math> form a Cauchy sequence. By the completeness of <math>X</math> there is a point <math>x\in X</math> such that <math>y_n\to x</math>. By the closure of each <math>C_n</math> and since <math>x</math> is in <math>C_n</math> for all <math>n\geq N</math>, <math>x\in\bigcap_{n=1}^\infty C_n</math>. To see that <math>x</math> is alone in <math>\bigcap_{n=1}^\infty C_n</math> assume otherwise. Take <math>x'\in\bigcap_{n=1}^\infty C_n</math> and then consider the distance between <math>x</math> and <math>x'</math> this is some value greater than 0 and implies that the <math>\lim_{n\to\infty} diam(C_n)\rightarrow d(x,x')>0</math>. Contradiction! Thus, <math>x</math> is very, very lonely in his small spartan little dorm room that is the infinite intersection of a sequence of closed set in a metric space that is complete, but how would he ever know.

==Statement for real numbers==

'''Theorem 2''':

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.

Suppose <math>X</math> is a compact topological space and <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots, \, </math>. Assume, by contradiction, that <math>\bigcap_{n=1}^\infty C_n=\varnothing</math>. Then we will build an open cover of <math>X</math> by considering the complement of <math>C_n</math> in <math>X</math>, i.e. <math>U_n=X\setminus C_n,\forall n</math>. Each <math>U_n</math> is open since the <math>C_n</math> are closed. Notice that <math>\bigcup_{n=1}^\infty U_n = \bigcup_{n=1}^\infty (X\setminus C_n) = X\setminus\bigcap_{n=1}^\infty C_n</math>, but we assumed that <math>\bigcap_{n=1}^\infty C_n=\varnothing</math> so that means <math>\bigcup_{n=1}^\infty U_n = X</math>. So, there are infinite many <math>U_n</math> covering our compact <math>X</math>. That means there exists a large <math>N</math> such that <math> X\subset\bigcup_{n=1}^N U_n</math>. Notice, however, that <math>C_n\supset C_{n+1},\forall n</math> implies that <math>X\setminus C_n=U_n\subset U_{n+1}=X\setminus C_{n+1},\forall n</math>. The only way for the nested and increasing <math>U_n</math> to cover <math>X</math> is if there is some index, call it <math>k</math>, such that <math>X=U_k</math>. This implies though that <math>C_k=X\setminus U_k=\varnothing</math>. This is a contradiction since we assumed that the <math>C_n</math> were non-empty. Hence, <math>\bigcap_{n=1}^\infty C_n\neq\varnothing</math>.

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.

Notice that in regards to the proof of Theorem 2, we don't need Hausdorffness. At no point in time do we appeal to the nature of points in the space. It is simply a statement about empty or not.

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.

Consider now a metric space <math>(X,d)</math> (not necessarily complete) in which <math>\bigcap_{n=1}^\infty C_n = x </math> whenever <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_n\supset C_{n+1},\forall n</math> and <math>\lim_{n\to\infty} diam(C_n)=sup\{d(x,y): x,y\in X\}\rightarrow 0</math>. Now, let <math>\{x_k\}</math> be a Cauchy sequence in <math> X</math> and take <math>C_n=\overline{\{x_k:k\geq n\}}</math>. The bar over the set means that we are taking the closure of the set under it. This guarantees that we are working with closed sets and since they contain the elements of our Cauchy sequence, we know them to be non-empty. In addition, <math>C_n\supset C_{n+1}</math> and since <math>\forall\epsilon>0,\exists N</math> such that when <math>n,m\geq N,d(x_n,x_m)<\epsilon</math>, (note this hold for all indices larger than our large <math>N</math>) then <math>diam(C_N)<\epsilon</math>. Hence, <math>\{C_n\}</math> satisfies the conditions above and there exists an <math>x\in X</math> such that <math>\bigcap_{n=1}^\infty C_n = x </math>. So, <math>x</math> is in the closure of all of the <math>C_n</math> and any open ball around <math>x</math> has non-empty intersection with the <math>C_n</math>. Now we will build a sub-sequence of the <math>\{x_n\}</math>, call it <math>\{x_{n_k}\}</math>, where <math>d(x,x_{n_k})<\frac{1}{k}</math>. This implies that <math>\{x_{n_k}\}\to x</math> and since <math>\{x_k\}</math> was Cauchy then it too must converge to <math>x</math>. Since <math>\{x_k\}</math> was an arbitrary Cauchy sequence, <math>X</math> is complete.

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

: <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''

:<math>C_0 \supset C_1 \supset \cdots C_n \supset C_{n+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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