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Date: 12-10-2018
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Date: 24-9-2019
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Date: 23-4-2019
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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
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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.
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ما أبرز التغيرات التي تحدث عند الرجال عندما يصبحون آباءً؟
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حقائق مثيرة للاهتمام حول الأرض
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إشادات بالملتقى الثّقافي التّربوي النّسوي الأوَّل
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