Content deleted Content added
Lim Wei Quan (talk | contribs) Corrected minor error |
Lim Wei Quan (talk | contribs) Corrected my missing backslashes and another typographical error |
||
Line 1:
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
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>.
| |||