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Alsosaid1987 (talk | contribs) →Topological Statement: compact does not imply closed in general topological spaces. need to add closed to theorem statement. |
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'''Cantor's intersection theorem''' refers to two closely related theorems in [[general topology]] and [[real analysis]], named after [[Georg Cantor]], about intersections of decreasing nested [[sequence|sequences]] of non-empty compact sets.
==Topological
'''Theorem.''' ''Let <math>S</math> be a [[topological space]]. A decreasing nested sequence of non-empty compact, closed subsets of <math>S</math> has a non-empty intersection. In other words, supposing <math>(C_k)_{k \geq 0}</math> is a sequence of non-empty compact, closed subsets of S satisfying''
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Since <math>C_0\subset S</math> is compact and <math>\{U_k \vert k \geq 0\}</math> is an open cover (on <math>C_0</math>) of <math>C_0</math>, a finite cover <math>\{U_{k_1}, U_{k_2}, \ldots, U_{k_m}\}</math> can be extracted. Let <math>M=\max_{1\leq i\leq m} {k_i}</math>. Then <math>{\textstyle \bigcup_{i = 1}^m U_{k_i}}=U_M</math> because <math>U_1\subset U_2\subset\cdots\subset U_n\subset U_{n+1}\cdots</math>, by the nesting hypothesis for the collection <math>(C_k)_{k \geq 0}</math>. Consequently, <math>C_0={\textstyle \bigcup_{i = 1}^m U_{k_i}} = U_M</math>. But then <math>C_M=C_0\setminus U_M=\emptyset</math>, a contradiction. [[Q.E.D.|∎]]
==Statement for
The theorem in real analysis draws the same conclusion for [[closed set|closed]] and [[bounded set|bounded]] subsets of the set of [[real number]]s <math>\mathbb{R}</math>. It states that a decreasing nested sequence <math>(C_k)_{k \geq 0}</math> of non-empty, closed and bounded subsets of <math>\mathbb{R}</math> has a non-empty intersection.
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