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. Theorem. "Let formula_1 be a topological space. A decreasing nested sequence of non-empty compact, closed subsets of formula_1 has a non-empty intersection. In other words, supposing formula_3 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 formula_1 is closed, for example when  formula_1 is Hausdorff. Proof. Assume, by way of contradiction, that formula_8. For each formula_9, let formula_10. Since formula_11 and formula_8, we have formula_13. Since the formula_14 are closed relative to formula_1 and therefore, also closed relative to formula_16, the formula_17, their set complements in formula_16, are open relative to formula_16. Since formula_20 is compact and formula_21 is an open cover (on formula_16) of formula_16, a finite cover formula_24 can be extracted. Let formula_25. Then formula_26 because formula_27, by the nesting hypothesis for the collection formula_3. Consequently, formula_29. But then formula_30, 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 formula_31. It states that a decreasing nested sequence formula_3 of non-empty, closed and bounded subsets of formula_31 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 formula_34, the intersection over formula_3  is formula_36. On the other hand, both the sequence of open bounded sets formula_37 and the sequence of unbounded closed sets formula_38 have empty intersection. All these sequences are properly nested. This version of the theorem generalizes to formula_39, the set of formula_40-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 formula_14 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" formula_3 "be a sequence of non-empty, closed, and bounded subsets of" formula_31 "satisfying" "Then," "Proof." Each nonempty, closed, and bounded subset formula_47 admits a minimal element formula_48. Since  for each formula_9, we have it follows that so formula_52 is an increasing sequence contained in the bounded set formula_16. The monotone convergence theorem for bounded sequences of real numbers now guarantees the existence of a limit point For fixed formula_9, formula_56 for all formula_57, and since formula_14 is closed and formula_59 is a limit point, it follows that formula_60. Our choice of formula_9 is arbitrary, hence formula_59 belongs to formula_63 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 formula_64  is a complete metric space, and formula_65 is a sequence of non-empty closed nested subsets of formula_64 whose diameters tend to zero:" "where formula_68 is defined by" "Then the intersection of the formula_14 contains exactly one point:" "for some formula_72." "Proof (sketch)." Since the diameters tend to zero, the diameter of the intersection of the formula_14 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 formula_74 for each formula_9. Since the diameter of formula_14 tends to zero and the formula_14 are  nested, the formula_48 form a Cauchy sequence. Since the metric space is complete this Cauchy sequence converges to some point formula_59. Since each formula_14 is closed, and formula_59 is a limit of a sequence in formula_14, formula_59 must lie in formula_14. This is true for every formula_9, and therefore the intersection of the formula_14 must contain formula_59. ∎ A converse to this theorem is also true: if formula_64 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 formula_64 is a complete  metric space. (To prove this, let formula_90 be a Cauchy sequence in formula_64, and let formula_14 be the closure of the tail formula_93 of this sequence.)