Cantor's intersection theorem

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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 sequences of non-empty compact sets.

Topological statement

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Theorem. Let   be a topological space. A decreasing nested sequence of non-empty compact, closed subsets of   has a non-empty intersection. In other words, supposing   is a sequence of non-empty compact, closed subsets of S satisfying

 

it follows that

 

The closedness condition may be omitted in situations where every compact subset of   is closed, for example when   is Hausdorff.

Proof. Assume, by way of contradiction, that  . For each  , let  . Since   and  , we have  . Since the   are closed relative to   and therefore, also closed relative to  , the  , their set complements in  , are open relative to  .

Since   is compact and   is an open cover (on  ) of  , a finite cover   can be extracted. Let  . Then   because  , by the nesting hypothesis for the collection  . Consequently,  . But then  , a contradiction.

Statement for real numbers

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The theorem in real analysis draws the same conclusion for closed and bounded subsets of the set of real numbers  . It states that a decreasing nested sequence   of non-empty, closed and bounded subsets of   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  , the intersection over   is  . On the other hand, both the sequence of open bounded sets   and the sequence of unbounded closed sets   have empty intersection. All these sequences are properly nested.

This version of the theorem generalizes to  , the set of  -element vectors of real numbers, but does not generalize to arbitrary metric spaces. For example, in the space of rational numbers, the sets

 

are closed and bounded, but their intersection is empty.

Note that this contradicts neither the topological statement, as the sets   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   be a sequence of non-empty, closed, and bounded subsets of   satisfying

 

Then,

 

Proof. Each nonempty, closed, and bounded subset   admits a minimal element  . Since for each  , we have

 ,

it follows that

 ,

so   is an increasing sequence contained in the bounded set  . The monotone convergence theorem for bounded sequences of real numbers now guarantees the existence of a limit point

 

For fixed  ,   for all  , and since   is closed and   is a limit point, it follows that  . Our choice of   is arbitrary, hence   belongs to   and the proof is complete. ∎

Variant in complete metric spaces

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

Theorem. Suppose that   is a complete metric space, and   is a sequence of non-empty closed nested subsets of   whose diameters tend to zero:

 

where   is defined by

 

Then the intersection of the   contains exactly one point:

 

for some  .

Proof (sketch). Since the diameters tend to zero, the diameter of the intersection of the   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   for each  . Since the diameter of   tends to zero and the   are nested, the   form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point  . Since each   is closed, and   is a limit of a sequence in  ,   must lie in  . This is true for every  , and therefore the intersection of the   must contain  . ∎

A converse to this theorem is also true: if   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   is a complete metric space. (To prove this, let   be a Cauchy sequence in  , and let   be the closure of the tail   of this sequence.)

See also

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References

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  • Weisstein, Eric W. "Cantor's Intersection Theorem". MathWorld.
  • Jonathan Lewin. An interactive introduction to mathematical analysis. Cambridge University Press. ISBN 0-521-01718-1. Section 7.8.