Cantor's intersection theorem

In real analysis, Cantor's intersection theorem is a theorem related to compact sets in R, the set of real numbers. It states that a decreasing nested sequence of non-empty, closed and bounded subsets of R has nonempty intersection. In other words, supposing {C_k} is a sequence of non-empty, closed and bounded sets satisfying

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

it follows that

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

The result is typically used as a lemma in proving the Heine-Borel theorem. Conversely, if the Heine-Borel theorem is known, then it can be restated as: a decreasing nested sequence of non-empty, compact subsets of R has nonempty intersection.

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.

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.

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

Consider the sequence (a_k) where a_k is the infimum over the non-empty C_k. Because C_k is closed, a_k belongs to C_k; because the sets are decreasing nested, the sequence is monotonic increasing. Because it is also bounded (being contained in the bounded set C₁), it must converge to some limit L. Choose any j ≥ 1; the subsequence of (a_k) for k ≥ j is contained in C_j and converges to L. Since j is closed, L lies in C_j. Since this is true for all j, L lies in all C_j, and so in their intersection.

References

• Weisstein, Eric W. "Cantor's Intersection Theorem". MathWorld.
• Jonathan Lewin. An interactive introduction to mathematical analysis. Cambridge University Press. ISBN 0521017181. Section 7.8.