5 fun facts about Cantor Set

cantor set photo

Photo by Det Springende Punkt

The first time that we heard about Cantor’ s name was in Chapter one of abbott’s text “Understanding Analysis”, which talks about Georg Cantor’s proof of uncountability of real numbers. This post explores 5 fun facts of a set that is named after him – the Cantor set.

Here is the construction of Cantor set:

Let C_{0} be the closed interval [0,1] and define C_{1} as the set that results when the open middle third of C_{0} is removed. C_{1} =C_{0} \backslash \left(\dfrac{1}{3},\dfrac{2}{3}\right).

Similarly, C_{2} is the set that results when the open middle third of each of the two intervals in C_{1} is removed. C_{2} =\left(\left[0,\dfrac{1}{9}\right]\cup\left[\dfrac{2}{9},\dfrac{1}{3}\right]\right)\cup\left(\left[\dfrac{2}{3},\dfrac{7}{9}\right]\cup\left[\dfrac{8}{9},1\right]\right).

Continue in this process and we can construct C_{n} consisting of 2^{n} closed intervals each having length of \dfrac{1}{3^{n}}. The Cantor set C is defined as C= \cap^{\infty}_{n=0}C_{n}

The picture above demonstrates the process of the construction of the Cantor set. The numbers represent the number of intervals remaining and the fractions represent the length of open intervals removed.

Fun Fact #1: Zero length

To form C_{1} , an open interval of length \dfrac{1}{3} was removed. Length of \dfrac{1}{3} in total was removed.

To form C_{2} , two open interval of length \dfrac{1}{9} was removed.Length of 2*\dfrac{1}{9} in total was removed.

To form C_{3} , four open interval of length \dfrac{1}{27} was removed.Length of 3*\dfrac{1}{27} in total was removed.

Continue in this fashion, to form C_{n} , 2^{n-1} open interval of length \dfrac{1}{3^{n}} was removed.2^{n-1}* \dfrac{1}{3^{n}} was removed.

Thus, the total length removed is

\sum^{\infty}_{n=0}\dfrac{1}{3}\left(\dfrac{2}{3}\right)^{n} = \dfrac{1}{3}+ 2*\dfrac{1}{9}+4*\dfrac{1}{27}+...+2^{n-1}*\dfrac{1}{3^{n}} +... = \dfrac{\dfrac{1}{3}}{1-\dfrac{2}{3}} = 1.

Since the total length removed is 1, the length of Cantor set is zero.

Fun Fact #2: Uncountable

It seems that the Cantor set is a very small and thin set, and intuitively we might think of it as a countable or even finite set. Surprisingly, the Cantor set is actually uncountable with cardinality equal to the cardinality of real numbers.

As we have proved in the first chapter of the book, the set [0,1] is uncountable. We can create a onto function f(x) that maps the Cantor set to the uncountable set [0,1]. According to the book Counterexamples in Topology, “If x\in C is written uniquely to the base 3 without using the digit 1, f(x) is the point in [0,1] whose binary expansion is obtained by replacing each digit ‘2’ in the ternary expansion of x by the digit ‘1’.” This onto function shows that the cardinality of the Cantor set is equal to or greater than the cardinality of real numbers. However, since the Cantor set is a subset of the real numbers, then cardinality of the Cantor set is equal to or less than the cardinality of real numbers. Thus, the cardinalities of the Cantor set and real numbers are the same. This result is really surprising, especially because the length of Cantor set is zero. If you are interested in this proof, you can read more here: https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf

Fun Fact #3: Fractal dimension

A point has a dimension of zero, a line has dimension one, a square has dimension two, and a cube has dimension three…

How will they change if we magnify each length by a 3?

A point undergoes no changes (3^{0}), a line triples in length (3^{1}), a square contains 9 copies of original square (3^{2}) and a cube contains 27 copies of original cube (3^{3}). The new copies is equal to the magnification factor to the power of dimension.

What is the dimension for Cantor set?

The Cantor set starts with C_{0} with the closed interval [0,1], and we want to magnify the Cantor set by starting with the closed interval [0,3]. Deleting open interval of the middle third of the closed interval [0,3], we will get [0,1] \cup [2,3], which is a starting point for exactly two Cantor set.

In this fashion, using a magnification factor of 3, we have two copies of the original set. Solve 2=3^{x}, we can get x(dimension) =\dfrac{log 2}{log 3} \simeq 0.631. Thus, the Cantor set has a fractal dimension of \dfrac{log 2}{log 3}.

Fun Fact #4: Closed 

One way to determine if a set C \in \mathbb{R} is closed is to check if it contains all of its limit points. It is also true that an intersection of closed sets is always closed. Because the Cantor set is the intersection of an arbitrary collection of closed sets, the Cantor set is closed.

Fun Fact #5: Compact

A set C \in \mathbb{R} is compact if every sequence in C has a subsequence that converges to a limit that is also in C. Also, Another characterization of compactness is that set C \in \mathbb{R} is compact if and only if it is closed and bounded. Since the Cantor set is constructed from closed interval [0,1], the Cantor set is bounded. Because the Cantor set is bounded and is closed as we proved in Fun Fact #4, it is compact!!

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Reference:

  • Abbott, Stephen. Understanding Analysis. New York: Springer, 2015. Print.
  • Steen, Lynn Arthur, and J. Seebach Arthur. Counterexamples in Topology. New York: Holt, Rinehart and Winston, 1970. Print.
  • Trinity College, The Cantor set is uncountable, Accessed on September 25, 2016, <https://www.maths.tcd.ie/~levene/221/pdf/cantor.pdf>

Posted on by Ruoping Shi
Categories: Fall 2016

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