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Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets

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Generalizations and Properties of the Ternary Cantor Set and Explorations in Similar Sets by Rebecca Stettin A capstone project submitted in partial fulfillment of graduating from the Academic Honors Program at Ashland University May 2017 Faculty Mentor: Dr. Darren D. Wick, Professor of Mathematics Additional Reader: Dr. Gordon Swain, Professor of Mathematics Abstract Georg Cantor was made famous by introducing the Cantor set in his works of mathemat- ics. This project focuses on different Cantor sets and their properties. The ternary Cantor set is the most well known of the Cantor sets, and can be best described by its construction. This set starts with the closed interval zero to one, and is constructed in iterations. The first iteration requires removing the middle third of this interval. The second iteration will remove the middle third of each of these two remaining intervals. These iterations continue in this fashion infinitely. Finally, the ternary Cantor set is described as the intersection of all of these intervals. This set is particularly interesting due to its unique properties being uncountable, closed, length of zero, and more. A more general Cantor set is created by tak- ing the intersection of iterations that remove any middle portion during each iteration. This project explores the ternary Cantor set, as well as variations in Cantor sets such as looking at different middle portions removed to create the sets. The project focuses on attempting to generalize the properties of these Cantor sets. i Contents Page 1 The Ternary Cantor Set 1 1 2 The n -ary Cantor Set 9 n−1 3 The n -ary Cantor Set 24 4 Conclusion 35 Bibliography 40 Biography 41 ii Chapter 1 The Ternary Cantor Set Georg Cantor, born in 1845, was best known for his discovery of the Cantor set. After writing a thesis in number theory, Cantor became interested in topology and wrote a series of papers on point set topology. He coined the term everywhere dense, which is currently used today. In this same series of papers, Cantor later introduced the idea of perfect sets. He made the ternary Cantor set famous in this same paper when attempting to prove that it was possible to have a set that is closed and nowhere dense. He noted that this was a perfect, infinite set that is nowhere dense in any interval, regardless of the size. (Fleron [3]) The Cantor set is created by starting with the closed interval [0,1]. The next step is to remove a middle portion of the interval. This initial removal will then leave two intervals, namely 1 2 [0, 3 ] and [ 3 , 1]. This process is continued an infinite number of times, with the final set of numbers, after the infinite iteration, being the Cantor set itself. This set is the intersection 1 th of all iterations, denoted as T = \i=1 Ti, where Ti represents the i iteration. There are 1 different proportions that can be removed with this set, the most well known being 3 . This set, referred to as the ternary Cantor set, can also be referred to as the classic set. This particular version of the Cantor set was the one Georg Cantor used to make his arguments about the nature of certain sets. (Aczel [1]) We will describe the construction of the ternary Cantor set, T , in a similar manner as it is 1 described in Thomson [7]. Begin with the closed interval [0; 1] and remove a dense open set, G. The remaining set, T = [0; 1] nG will also be closed and nowhere dense in [0; 1]. Based on our construction of G, T will have no isolated points. It is easiest to understand the set 1 2 G if we construct it in stages. Let G1 = ( 3 ; 3 ) and let T1 be what is remaining in [0; 1] after 1 2 removing G1. Thus T1 = [0, 3 ] [ [ 3 , 1] is what remains when the middle third of the interval [0,1] is removed. This is referred to as the first iteration of the ternary Cantor set. We repeat 1 2 7 8 this construction for each of the two intervals of T1. Let G2 = ( 9 ; 9 ) [ ( 9 ; 9 ). These intervals 1 2 1 2 7 8 are the middle third of the previous 2 intervals. Then T2 = [0; 9 ] [ [ 9 ; 3 ] [ [ 3 ; 9 ] [ [ 9 ; 1]. This completes the second iteration. We continue inductively, and ultimately take the intersection of each iteration of Ti to create the ternary Cantor set with the following properties: For each i 2 N i 1. Ti is a union of 2 pairwise disjoint closed intervals. 2. Each component of the set of open intervals removed is the "middle third" of some component of Ti. 1 3. The length of each component of Ti is 3i . Some proofs of the properties of the ternary Cantor set will be explored in later chapters in regards to more generalized Cantor sets. The following is a list of characteristics of the ternary Cantor set with some proofs to follow: 1. The Cantor set is nonempty. 2. The Cantor set is closed. 3. The Cantor set has length 0. 4. The Cantor set contains all numbers that can be written in base-3 without 1's. 5. The Cantor set is uncountable. 6. The Cantor set is nowhere dense. 2 7. The Cantor set has no isolated points. 8. The Cantor set is a perfect set. Theorem 1. The ternary Cantor set is nonempty. Proof. Using similar methods as Nelson [5], we must show that the ternary Cantor set contains at least one element. We can see that during each iteration, the endpoints remain 1 2 in the set. For example, during T1, everything between ( 3 , 3 ) is removed. Note that it is everything between the two endpoints, which implies the two endpoints remain. Once 1 2 removed, T1 is [0, 3 ] [ [ 3 , 1]. As seen in here, the intervals are closed, thus including the endpoints. This trend continues throughout each iteration. For T2, the middle thirds of 1 those two previously mentioned intervals are removed. To clarify, everything between ( 9 , 2 7 8 1 2 1 2 7 8 9 ) as well as ( 9 , 9 ) is removed. This leaves [0, 9 ] [ [ 9 , 3 ] [ [ 3 , 9 ] [ [ 9 , 1], which is once 1 2 again a union of closed intervals, which include endpoints. Notice that 0, 3 , 3 , and 1 are again included in this iteration. These iterations continue infinitely. Since each iteration of T contains the endpoints of each iteration before it, it becomes clear that the endpoints of each iteration are included. Since the previous endpoints are consistently included, there will clearly be elements in the intersection of each iteration, proving the ternary Cantor set is nonempty. Theorem 2. The ternary Cantor set is closed. Proof. As shown earlier when looking at the length of T, it is an intersection of closed sets. 1 2 For example, T1 is [0, 3 ] [ [ 3 , 1]. This trend continues with each Ti also consisting of a finite union of closed intervals. Each Ti is closed since it is a finite union of closed intervals. 1 Since T = \i=1Ti it is a countable intersection of closed sets and is therefore closed. Theorem 3. The ternary Cantor set has length of 0. Proof. This proof is also based off of similar methods used in Nelson [5]. This proof can be demonstrated by noting everything that is taken out of the starting interval has the same 3 length as the initial interval. Looking at what is taken out during each iteration, a sequence 1 emerges. During the first iteration, an interval of length 3 is removed from the set. During 1 2 the second iteration, there are 2 different intervals, namely [0, 3 ] and [ 3 ,1], to remove the 1 1 1 middle third from. In this case, however, the length of what is removed is 3 of 3 , or 9 . This 1 th must occur twice because there are 2 intervals that ( 9 ) is being removed from. Therefore, 2 2 a length of 32 = 9 will be removed during the second iteration of T. This trend continues 1 2 4 8 throughout each iteration resulting in the length removed at each iteration as 3 , 9 , 27 , 81 ,··· The total length removed can also be represented as 1 X 2i 3i+1 i=0 1 1 This sum can be determined through the sum as a geometric series, or 3 2 = 1. This 1− 3 means that everything removed has length 1. As mentioned previously, the starting interval, T0, has length 1 as well. A simple subtraction of the removed portion from the initial length will give the final length of T as 0. Therefore, T has length 0. The following theorem will be explored through examples, but will not be proven. A similar proof can be found in Chapter 3. Theorem 4. The ternary Cantor set only contains numbers that have a base-3 representation excluding the digit 1. A brief review of base-3 representation is necessary to proceed with this example. Base-3, or ternary, representation consists of 3 digits, 0, 1, and 2. A number of the form (.a1a2a3··· )3 n can be thought of in terms of sums of fractions with numerators an and denominators 3 . a1 a2 a3 That is, (.a1a2a3··· )3 = 3 + 32 + 33 + ··· where an must be either 0, 1, or 2. The claim is every number that is in the ternary Cantor set can be written in base-3 representation with only 2's and 0's.
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