# The Cantor Set

By Nick Wan and Elaine Wissuchek

**Introduction to the Cantor Set**

Cantor set is a special subset of the closed interval [0, 1] invented by a German mathematician Georg Cantor in 1883. In order to construct this set, we need to construct infinitely many subset of inductively and take the intersection of all of them. Specifically:

Let . Remove the open middle third from .

Let , precisely . Remove middle third open intervals and from the respective closed intervals whose union is .

Let say, .

Continue this procedure of removal of open intervals from each closed interval where whose union is where and get by taking the union of whatever is left from each after the removal of open intervals as before.

Define the Cantor set .

Another way to view the Cantor set is in terms of ternary expansions:

Given a real number, there is a sequence of integers with such that the series converges to . In other words, we can write in a ternary form: (base 3). For example, .

In the case or our Cantor set, any point in has the following ternary expansion:

with all from the set .

For example, the element of the Cantor set corresponds to and corresponds to .

**Uncountability of the Cantor Set by Diagonalization**

We can use Cantor’s diagonalization method to show that the Cantor set is not countable.

Let be a sequence so that each represents an element of the Cantor set. Let all . Assume that is a one-to-one correspondence between the natural numbers and the set so that the elements of the cantor set are countable.

Consider where if and if for and . Therefore will be distinct from any in the th position. Thus for , we cannot find an for which , so is not onto, a contradiction to the one-to-one correspondence of . Hence by contradicting the definition of countability, that , is uncountable. Since all elements of the cantor set are represented in , the Cantor set is uncountable.

**The Cantor Set is Closed **

Each is a union of closed intervals. As a finite union of closed intervals, it is a closed set in . Then is a closed subset of being the intersection of closed sets for

**The Cantor Set is perfect**

A set is considered perfect if the set is closed and all the points of the set are limit points of the set. For each endpoint in the set there will always exist another point in the set within a deleted neighborhood of some radius on one side of that point since the remaining intervals at each step are being divided into infinitely small subintervals and since the real numbers are infinitely dense. Likewise, for each nonendpoint in the set there will always exist another point in the set within a deleted neighborhood of some radius on both sides of that point. Hence, all deleted neighborhoods of any radius around each point of the set for which the intersection of that deleted neighborhood and the set are nonempty. Therefore, each point in the set is a limit point of the set, and since the set is closed, the set is perfect.

**Dimension of the Cantor Set**

Definition: Dimension

- The Topological Definition: A set is of dimension when each point has an neighborhood whose boundaries meet other points in a set of dimension and is the lease non-negative integer for which this holds.

The empty set has a topological dimension () of -1. Consider a finite set of points because by the definition of isolated points, they are not limit points, so there exists an neighborhood that does not intersect . Instead, this neighborhood intersects the empty set, so a finite set of points will have .

Notice that we can choose for any so that there exists an neighborhood that does not intersect the Cantor set. Instead, this neighborhood intersects the empty set, so the Cantor set’s topological dimension () is 0.

However, this does not account for the infinite number of points that will still exist within the defined epsilon neighborhood of the point in the cantor set, wherever . The Topological definition of dimension is further limited because it does not describe the way the cantor set scales well. To fix these problems, we turn to a definition that allows for fractional dimensions.

- The Hausdorff-Besicovitch Definition: The exponent that a scale factor must take so that where is the number of scaled objects needed to create the original object.

When the Cantor set is scaled to of its original size, it takes of the scaled objects to recreate the original Cantor set. Thus . The Cantor set’s HB dimension () .

Not only is a fraction, , whereas typically when .

References:

http://www.ias.ac.in/article/fulltext/reso/019/11/1000-1004

http://www.iiserpune.ac.in/~supriya/teaching/Topology-MTH322/files/CantorSet.pdf

http://studylib.net/doc/11695972/an-exploration-of-the-cantor-set-introduction

http://blog.mathteachersresource.com/?p=848

https://wakespace.lib.wfu.edu/bitstream/handle/10339/39274/Walsh_wfu_0248M_10559.pdf

Good introduction to the Cantor Set and fundamental properties thereof. There is a bit of confusion in notation in the section of Cantor’s diagonalization argument. Readers should be cautious as they read this section, and see if they can identify the notational issues there.