The Cantor set is a subset of the real numbers that exhibits a number of interesting and counter-intuitive properties. It is among the simplest examples of a fractal. Topologically, it is a closed set, and also a perfect set. Despite containing an uncountable number of elements, it has Lebesgue measure equal to .
The Cantor set can be described recursively as follows: begin with the closed interval , and then remove the open middle third segment , dividing the interval into two intervals of length . Then remove the middle third of the two remaining segments, and remove the middle third of the four remaining segments, and so on ad infinitum.
Another equivalent representation for is: Start with the interval , then scale it by . Then join it with a copy shifted by , and repeat ad infinitum.
Using this representation, can be rendered in LaTeX:
A distorted version of can be found by repeatedly applying the function , and keeping the values of x for which the values always remain bounded. This constructs by repeatedly removing the middle. This works since for the values will always diverge, and the values of for which is the union of intervals , which are disjoint when . -- EVIN-
This article is a stub. Help us out by.