Cantor's intersection theorem: Difference between revisions

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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 C_n\}\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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Notice the differences and the similarities between the two theorem. In Theorem 2, the <math>C_n</math> are only assumed to be closed (and not compact, which is stronger) since given a compact space <math>X</math> and <math>Y\subset X</math> a closed subset, <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 having the Cantor Intersection property (i.e. the theorem above holds) is necessarily 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>

Notice that each of the hypotheses above is essential. If the metric space were not complete, then one could construct a nested sequence of non-empty, compact sets converging to a "hole" in the space, i.e. <math>\mathbb{Q}</math> with the usual metric and the sequence of sets, <math>C_n = [\sqrt{2}, \sqrt{2}+1/n]</math>. If the sets are not closed, then one can construct sequences of nested sets which have empty intersection, i.e. <math>\mathbb{R}</math> with the collection, <math>C_n = (0-\frac{1}{n},0+\frac{1}{n}) </math> or the collection <math>C_n = [n, \infty)</math>. This last case also demonstrates what can happen if the diameters are not tending to 0. However, another example would be <math>C_n = [0-\frac{1}{n}, 1+\frac{1}{n}]</math> and the infinite intersection will yield more than a single point.

== Proof ==