Cantor Set - Composition

Composition

Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit interval remaining can be found by total length removed. This total is the geometric progression

So that the proportion left is 1 – 1 = 0.

This calculation shows that the Cantor set cannot contain any interval of non-zero length. In fact, it may seem surprising that there should be anything left — after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment (1/₃, 2/₃) from the original interval leaves behind the points 1/₃ and 2/₃. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an infinite number of points.

It may appear that only the endpoints are left, but that is not the case either. The number 1/4, for example, is in the bottom third, so it is not removed at the first step, and is in the top third of the bottom third, and is in the bottom third of that, and in the top third of that, and so on ad infinitum—alternating between top and bottom thirds. Since it is never in one of the middle thirds, it is never removed, and yet it is also not one of the endpoints of any middle third. The number 3/10 is also in the Cantor set and is not an endpoint.

In the sense of cardinality, most members of the Cantor set are not endpoints of deleted intervals.