Cantor's intersection theorem, attributed to Georg Cantor (Moreno et al. 2020, p. 41), states that a decreasing sequence
of bounded nonempty closed sets
in the real numbers has a point intersection , for all dimensions
of Euclidean space .
Note that the hypotheses stated above are crucial. The infinite intersection of open intervals
may be empty , for instance 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 .
See also Bolzano's Theorem ,
Bolzano-Weierstrass Theorem ,
Bounded Set ,
Cantor
Set ,
Closed Set ,
Compact
Set ,
Heine-Borel Theorem ,
Intersection ,
Real Number ,
Topological
Space
Portions of this entry contributed by Todd
Rowland
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References Moreno, J.; Blanco, M.; and Pascual, P. "La compacitat, una noció seminal en l'evolució de la topologia general (1895-1930)."
Butlletí de la Societat Catalana de Matemàtiques 35 ,
23-56, 2020. https://doi.org/10.2436/20.2002.01.92 . Referenced
on Wolfram|Alpha Cantor's Intersection Theorem
Cite this as:
Weisstein, Eric W. , with contributions by Todd Rowland . "Cantor's Intersection Theorem." From MathWorld https://mathworld.wolfram.com/CantorsIntersectionTheorem.html
Subject classifications
in their intersection, 
for all 
. For example, ![0 in intersection [0,1/n]](/images/equations/CantorsIntersectionTheorem/Inline4.svg)
. It is also true in higher dimensions
of Euclidean space.
. Also, the infinite intersection of unbounded closed
sets may be empty, e.g., 
.
