Cantor's intersection theorem

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

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

C_{0}\supseteq C_{1}\supseteq \cdots C_{k}\supseteq C_{k+1}\cdots ,

it follows that

\left(\bigcap _{k}C_{k}\right)\neq \emptyset .

Proof

Assume, by way of contradiction, that $\bigcap C_{n}=\emptyset$ . For each n, let $U_{n}=C_{0}\setminus C_{n}$ . Since $\bigcup U_{n}=C_{0}\setminus \bigcap C_{n}$ and $\bigcap C_{n}=\emptyset$ , thus $\bigcup U_{n}=C_{0}$ .

Since $C_{0}$ ⊂ $S$ is compact and $(U_{n})$ is an open cover of $C_{0}$ , we can extract a finite cover $\{U_{n_{1}},U_{n_{2}},\ldots ,U_{n_{m}}\}$ . Let $U_{k}$ be the largest set of this cover; then $C_{0}$ ⊂ $U_{k}$ . But then $C_{k}=C_{0}\setminus U_{k}=\emptyset$ , 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) 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} 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, ∞) have empty intersection. All these sequences are properly nested.

This version of the theorem generalizes to Rⁿ, 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 sets

C_{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.

Proof

Each closed bounded non-empty subset C_k of R admits a minimal element x_k. Since for each k, we have

x_{k+1}\in C_{k+1}\subseteq C_{k}

it follows that

x_{k}\leq x_{k+1}

so (x_k) is an increasing sequence contained in the bounded set C₀. The Monotone convergence theorem now guarantees the existence of a limit point

x=\lim _{k\to \infty }x_{k}.

For fixed k, we have that x_j∈C_k for all j≥k and since C_k was closed, it follows that x∈C_k. Our choice of k was arbitrary, hence x belongs to the intersection of all 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. Suppose that X is a non-empty complete metric space, and C_n is a sequence of closed nested subsets of X whose diameters tend to zero:

\lim _{n\to \infty }\operatorname {diam} (C_{n})=0

where diam(C_n) is defined by

\operatorname {diam} (C_{n})=\sup\{d(x,y)|x,y\in C_{n}\}.

Then the intersection of the C_n contains exactly one point:

\bigcap _{n=1}^{\infty }C_{n}=\{x\}

for some x in X.

Proof

A proof goes as follows. Since the diameters tend to zero, the diameter of the intersection of the C_n 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_n of C_n for each n. Since the diameter of C_n tends to zero and the C_n are nested, the x_n form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point x. Since each C_n is closed, and x is a limit of a sequence in C_n, x must lie in C_n. This is true for every n, and therefore the intersection of the C_n 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 closed subsets whose diameters tend to zero is non-empty, then X is a complete metric space. (To prove this, let x_n be a Cauchy sequence in X, and let C_n be the closure of the tail of this sequence.)

References

Weisstein, Eric Wolfgang. "Cantor's Intersection Theorem". MathWorld.

Jonathan Lewin. An interactive introduction to mathematical analysis. Cambridge University Press. ISBN 0-521-01718-1. Section 7.8.

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