Cantor's intersection theorem explained

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

Theorem. Let

S be a topological space. A decreasing nested sequence of non-empty compact, closed subsets of S has a non-empty intersection. In other words, supposing (C _{k)} _{k} is a sequence of non-empty compact, closed subsets of S satisfyingC _{0} \supset C _{1} \supset … \supset C _{n} \supset C _{n+1} \supset … ,

it follows that

c ap ^{inf ty} _{k =0} C _{k} ≠ \emptyset.

The closedness condition may be omitted in situations where every compact subset of

S is closed, for example when S is Hausdorff.

Proof. Assume, by way of contradiction, that

{ style c ap ^{inf ty} _{k =0} C _{k}=\emptyset} . For each k , let U _{k=C} _{0\setminus} C _{k} . Since { style c up ^{inf ty} _{k =0} U _{k}=C} _{0\setminus} { style c ap ^{inf ty} _{k =0} C _{k}} and { style c ap ^{inf ty} _{k =0} C _{k}=\emptyset} , we have { style c up ^{inf ty} _{k =0} U _{k}=C} _{0} . Since the C _{k} are closed relative to S and therefore, also closed relative to C _{0} , the U _{k} , their set complements in C _{0} , are open relative to C _{0} .

Since

C _{0\subset} S is compact and \{U _{k} \vert k \geq 0 \} is an open cover (on C _{0} ) of C _{0} , a finite cover \{U _{k _{1}} , U _{k _{2}} , \ldots, U _{k _{m}} \} can be extracted. Let M =max _{1\leq} {k _{i}} . Then { style c up ^{m} _{i =1} U _{k _{i}} }=U_M because U _{1\subset} U _{2\subset … \subset} U _{n\subset} U _{n+1} … , by the nesting hypothesis for the collection (C _{k)} _{k} . Consequently, C _{0={stylecup} ^{m} ^{i =1} U _{k _{i}} } = U_M. But then C _{M=C} _{0\setminus} U _{M=\emptyset} , a contradiction. ∎

Statement for real numbers

The theorem in real analysis draws the same conclusion for closed and bounded subsets of the set of real numbers

R . It states that a decreasing nested sequence (C _{k)} _{k} of non-empty, closed and bounded subsets of R 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

C _{k=[0,1/k]} , the intersection over (C _{k)} _{k} is \{ 0 \} . On the other hand, both the sequence of open bounded sets C _{k=(0,1/k)} and the sequence of unbounded closed sets C _{k=[k,infty)} have empty intersection. All these sequences are properly nested.

This version of the theorem generalizes to

R ^{n} , the set of n -element vectors of real numbers, but does not generalize to arbitrary metric spaces. For example, in the space of rational numbers, the setsC _{k} = [\sqrt{ 2 }, \sqrt{ 2 } +1 /k] = (\sqrt{ 2 }, \sqrt{ 2 } +1 /k)

are closed and bounded, but their intersection is empty.

Note that this contradicts neither the topological statement, as the sets

C _{k} 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

(C _{k)} _{k} be a sequence of non-empty, closed, and bounded subsets of R satisfyingC _{0} \supset C _{1} \supset … C _{n} \supset C _{n+1} … .

Then,

c ap ^{inf ty} _{k =0} C _{k} ≠ \emptyset.

Proof. Each nonempty, closed, and bounded subset

C _{k\subsetR} admits a minimal element x _{k} . Since for each k , we have x _{k+1} \in C _{k+1} \subset C _{k} ,it follows thatx _{k} \le x _{k+1} ,

so

(x _{k)} _{k} is an increasing sequence contained in the bounded set C _{0} . The monotone convergence theorem for bounded sequences of real numbers now guarantees the existence of a limit pointx = \lim _{k\to} x _{k.}

For fixed

k , x _{j\in} C _{k} for all j\geq k , and since C _{k} is closed and x is a limit point, it follows that x\in C _{k} . Our choice of k is arbitrary, hence x belongs to { style c ap ^{inf ty} _{k =0} C _{k}} 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

X is a complete metric space, and (C _{k)} _{k} is a sequence of non-empty closed nested subsets of X whose diameters tend to zero:\lim _{k\toinfty} \operatorname{diam}(C _{k)} = 0 ,

where

\operatorname{diam}(C _{k)} is defined by\operatorname{diam}(C _{k)} = \sup\{d(x,y) \mid x,y\in C _{k\}.}

Then the intersection of the

C _{k} contains exactly one point:c ap ^{inf ty} _{k =1} C _{k} = \{x\}

for some

x \in X .

Proof (sketch). Since the diameters tend to zero, the diameter of the intersection of the

C _{k} 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 x _{k\in} C _{k} for each k . Since the diameter of C _{k} tends to zero and the C _{k} are nested, the x _{k} form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point x . Since each C _{k} is closed, and x is a limit of a sequence in C _{k} , x must lie in C _{k} . This is true for every k , and therefore the intersection of the C _{k} must contain x . ∎

A converse to this theorem is also true: if

X 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 X is a complete metric space. (To prove this, let (x _{k)} _{k} be a Cauchy sequence in X , and let C _{k} be the closure of the tail (x _{j)} _{j} of this sequence.)

See also

• Kuratowski's intersection theorem
• Helly's theorem - another theorem on intersection of sets.

References

• • Jonathan Lewin. An interactive introduction to mathematical analysis. Cambridge University Press. . Section 7.8.