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# 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.

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

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## Cite this as:

Rowland, Todd. "Cantor's Intersection Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CantorsIntersectionTheorem.html