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., .