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The typesetting is ugly as sin, but I know latex not whatever this markup language is. Everything is correct now and there are a few more references. As a result I am removing the flag at the top. The statement of both theorems are correct. |
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In [[real analysis]], a branch of mathematics, '''Cantor's intersection theorem''', named after [[Georg Cantor]], gives conditions under which an infinite intersection of nested, non-empty, sets is non-empty.
'''Theorem 1''': If <math>(X, d)</math> is a non-trivial, complete, metric space and <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_n\supset C_{n+1},\forall n</math> and <math>\lim_{n\to\infty} diam(C_n)=sup\{d(x,y): x,y\in X\}\rightarrow 0</math>. Then, there exists an <math>x\in X</math> such that <math>\bigcap_{n=1}^\infty C_n = x </math> <ref>"Real Analysis," H.L. Royden, P.M. Fitzpatrick, 4th edition, 2010, page 195</ref>.
'''Theorem 2''': If <math>X</math> is a compact space and <math>\{C_n\}</math> is an infinite sequence of non-empty, closed sets such that <math>C_1 \supseteq \cdots C_k \supseteq C_{k+1} \cdots, \, </math>, then <math>\bigcap_{n=1}^\infty C_n\neq\varnothing</math>.
Notice the differences and the similarities between the two theorem. In Theorem 2, the <math>C_n</math> are only required to be closed since given a compact space <math>X</math> and <math>Y\subset X</math> a closed subset, then <math>Y</math> is necessarily compact. Also, in Theorem 1 the intersection is exactly 1 point, while in Theorem 2 it could contain many more points. Interestingly, a metric space can only have the Cantor Intersection property (i.e. the theorem above holds) if it is complete (for justification see below). An example of an application of this theorem is the existence of limit points for self-similar contracting fractals<ref>Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces, T. Bedford, M. Keane and C. Series eds., Oxford Univ. Press 1991, page 225</ref>.
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