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.
A decreasing nested sequence of non-empty compact, closed subsets of
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
∎ 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
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.
On the other hand, both the sequence of open bounded sets
and the sequence of unbounded closed sets
This version of the theorem generalizes to
-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.
be a sequence of non-empty, closed, and bounded subsets of
Each nonempty, closed, and bounded subset
admits a minimal element
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
∎ In a complete metric space, the following variant of Cantor's intersection theorem holds.
is a complete metric space, and
is a sequence of non-empty closed nested subsets of
is zero, so it is either empty or consists of a single point.
So it is sufficient to show that it is not empty.
Since the metric space is complete this Cauchy sequence converges to some point
∎ 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