Cantor,s Intersection Theorem
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المرجع الالكتروني للمعلوماتيه
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المرجع الالكتروني للمعلوماتيه
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19-9-2018
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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, ![0 in intersection [0,1/n]](http://mathworld.wolfram.com/images/equations/CantorsIntersectionTheorem/Inline4.gif)
. 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., ![intersection [n,infty]](http://mathworld.wolfram.com/images/equations/CantorsIntersectionTheorem/Inline6.gif)
.
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.
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