The Cantor Set
The Cantor set, introduced by German mathematician Georg Cantor, is a construction of a set of points lying on a single line segment, and involving only the real numbers between zero and one.
The Cantor set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third from the interval [0, 1], leaving two line segments: [0, 1/3] U [2/3, 1]. Next, the open middle third of each of these remaining segments is deleted. This process is continued ad infinitum. The Cantor set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process.
It can be shown that there are as many points left behind in this process as there were that were removed, and that therefore, the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set C to the closed interval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1]. Since C is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal.
