Cantor's intersection theorem: Difference between revisions

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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.

{{Disputeabout|Statement of theorem|date=May 2013}}

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\to\infty} diam(C_n)=sup\{d(x,y): x,y\in X\}\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>.

In [[real analysis]], a branch of mathematics, '''Cantor's intersection theorem''', named after [[Georg Cantor]], is a theorem related to [[compact set]]s of a compact space <math>S</math>. It states that a decreasing nested [[sequence]] of non-empty compact subsets of <math>S</math> has nonempty intersection. In other words, supposing {''C''_''k''} is a sequence of non-empty, closed and bounded sets satisfying

Theorem 2: If <math>C_0X</math> is a compact space and <math>\supseteq{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>, then <math>\bigcap_{n=1}^\infty C_n\neq\varnothing</math>.

Notice the differences and the similarities between the two theorem. In Theorem 2, the <math>C_n</math> are only required to be closed since given a compact space <math>X</math> and <math>Y\subset X</math> a closed subset, then <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 can only have the Cantor Intersection property (i.e. the theorem above holds) if it is 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>.

it follows that

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},0+\frac{1}{n}) </math> or the collection <math>C_n = [n, \infty)</math>. This last case also demonstrates what can happen if the diameters are not tending to 0. However, another example would be <math>C_n = [0-\frac{1}{n}, 1+\frac{1}{n}]</math> and the infinite intersection will yield more than a single point. Violation of the remaining hypotheses are clear and left to the reader as an exercise (I have always wanted to write that).

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

== Proof ==

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 a compact space has nonempty intersection.

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.

Theorem 2:

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.

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 compliment 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> since I am throwing out less and less stuff each time. 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>.

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.

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

: <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.

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.

== Proof ==

Suppose that <math>\bigcap C_n=\emptyset</math>. Let <math>U_n=X\setminus C_n</math>. Since <math>\bigcup U_n=X\setminus\bigcap C_n</math> and <math>\bigcap C_n=\emptyset</math>, then <math>\bigcup U_n=X</math>.

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.

Since <math>X</math> is compact and <math>(U_n)</math> is an open cover of it, we can extract a finite cover. Let <math>U_k</math> be the largest set of this cover, then <math>\bigcap C_n=C_k\neq\emptyset</math> by hypothesis.

== 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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