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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_n\supset C_{n+1},\forall n</math>, then <math>\bigcap_{n=1}^\infty C_n\neq\varnothing</math>.
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