Cantor's intersection theorem

In real analysis, a branch of mathematics, Cantor's intersection theorem, named after Georg Cantor, is a theorem related to compact sets of a compact space ${\displaystyle S}$. It states that a decreasing nested sequence of non-empty compact subsets of ${\displaystyle S}$ has nonempty intersection. In other words, supposing {C_k} is a sequence of non-empty, closed and bounded sets satisfying

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

it follows that

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

The result is typically used as a lemma in proving the Heine–Borel theorem, which states that sets of real numbers are compact if and only if they are closed and bounded. Conversely, if the Heine–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.

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

${\displaystyle C_{k}=[{\sqrt {2}},{\sqrt {2}}+1/k]=({\sqrt {2}},{\sqrt {2}}+1/k)}$

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

Suppose that ${\displaystyle \bigcap C_{n}=\emptyset }$ . Let ${\displaystyle U_{n}=X\setminus C_{n}}$ . Since ${\displaystyle \bigcup U_{n}=X\setminus \bigcap C_{n}}$  and ${\displaystyle \bigcap C_{n}=\emptyset }$ , then ${\displaystyle \bigcup U_{n}=X}$ .

Since ${\displaystyle X}$  is compact and ${\displaystyle (U_{n})}$  is an open cover of it, we can extract a finite cover. Let ${\displaystyle U_{k}}$  be the largest set of this cover, then ${\displaystyle \bigcap C_{n}=C_{k}\neq \emptyset }$  by hypothesis.

References

• 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.