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

 C_1 superset C_2 superset C_3 superset ...

in the real numbers, then Cantor's intersection theorem states that there must exist a point p in their intersection, p in C_n for all n. For example, 0 in  intersection [0,1/n]. 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  intersection (0,1/n). Also, the infinite intersection of unbounded closed sets may be empty, e.g.,  intersection [n,infty].

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.

See also

Bolzano's Theorem, Bolzano-Weierstrass Theorem, Bounded Set, Cantor Set, Closed Set, Compact Set, Heine-Borel Theorem, Intersection, Real Number, Topological Space

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Cantor's Intersection Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications