Cantor set—an infinite set of numbers between 0 and 1, defined by an inductive process. To define this set, start with the closed interval [0,1]. Remove the middle third--the open interval (1/3,2/3). (That is, remove all the points between 1/3 and 2/3, except the points 1/3 and 2/3 themselves.) Next take each of the remaining two intervals, and remove its middle third. This leaves us with four intervals: [0,1/9], [2/9,1/3], [2/3,7/9], and [8/9,1]. Now remove the middle third of each of these intervals. This procedure is repeated ad infinitum. The resulting set is the Cantor set, named for Georg Cantor.

This set has many interesting properties. It is an example of a set of measure 0 (see Measurable and nonmeasurable). This is because the interval [0,1] has length 1, and if we add up the lengths of all the removed middle thirds, we also get 1. The Cantor set nevertheless contains infinitely many points--in fact, it contains many points. Moreover...