Cantor's Intersection Theorem

A theorem about (or providing an equivalent definition of) compact sets, originally due to Georg Cantor. Given a decreasing sequence of bounded nonempty closed sets

in the real numbers, then Cantor's intersection theorem states that there must exist a point  in their intersection,  for all . For example, . It is also true in higher dimensions of Euclidean space.

Note that the hypotheses stated above are crucial. The infinite intersection of open intervals may be empty, for instance . Also, the infinite intersection of unbounded closed sets may be empty, e.g., .

Cantor's intersection theorem is closely related to the Heine-Borel theorem and Bolzano-Weierstrass theorem, each of which can be easily derived from either of the other two. It can be used to show that the Cantor set is nonempty.