Cantor set

In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883^[1] (but discovered in 1875 by Henry John Stephen Smith ^[2]), is a set of points lying on a single line segment that has a number of remarkable and deep properties. Through consideration of it, Cantor and others helped lay the foundations of modern general topology. Although Cantor himself defined the set in a general, abstract way, the most common modern construction is the Cantor ternary set, built by removing the middle thirds of a line segment. Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

Contents 1 Construction of the ternary set 2 What's in the Cantor set? 3 Properties 3.1 Cardinality 3.2 Self-similarity 3.3 Topological and analytical properties 4 Variants of the Cantor set 5 Historical remarks 6 See also 7 References

Construction of the ternary set

The Cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. One starts by deleting the open middle third (¹/₃, ²/₃) from the interval [0, 1], leaving two line segments: [0, ¹/₃] ∪ [²/₃, 1]. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [0, ¹/₉] ∪ [²/₉, ¹/₃] ∪ [²/₃, ⁷/₉] ∪ [⁸/₉, 1]. This process is continued ad infinitum. The Cantor ternary set contains all points in the interval [0, 1] that are not deleted at any step in this infinite process. The first six steps of this process are illustrated below.

What's in the Cantor set?

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

<math>\sum_{n=0}^\infty \frac{2^n}{3^{n+1}} = \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \cdots = \frac{1}{3}\left(\frac{1}{1-\frac{2}{3}}\right) = 1.</math>

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 [0, 1] 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 ¼, 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.

Properties

Cardinality

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. To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation. In this notation, ¹/₃ can be written as 0.1₃ and ²/₃ can be written as 0.2₃, so the middle third (to be removed) contains the numbers with ternary numerals of the form 0.1xxxxx...₃ where xxxxx...₃ is strictly between 00000...₃ and 22222...₃. So the numbers remaining after the first step consists of

• Numbers of the form 0.0xxxxx...₃
• ¹/₃ = 0.1₃ = 0.022222...₃ (This alternative recurring representation of a number with a terminating numeral occurs in any positional system.)
• ²/₃ = 0.122222...₃ = 0.2₃
• Numbers of the form 0.2xxxxx...₃

All of which can be stated as those numbers with a ternary numeral 0.0xxxxx...₃ or 0.2xxxxx...₃ The second step removes numbers of the form 0.01xxxx...₃ and 0.21xxxx...₃, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral whose first two digits are not 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must have a numeral consisting entirely of 0s and 2s. It is worth emphasising that numbers like 1, ¹/₃ = 0.1₃ and ⁷/₉ = 0.21₃ are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.2222...₃, ¹/₃ = 0.022222...₃ and ⁷/₉ = 0.2022222...₃. So while a number in C may have either a terminating or a recurring ternary numeral, one of its representations will consist entirely of 0s and 2s. It has been conjectured that all algebraic irrational numbers are normal and if true, this would imply that all members of the Cantor set are either rational or transcendental. The function from C to [0,1] is defined by taking the numeral that does consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number. In a formula,

<math>f \left( \sum_{k=1}^\infty a_k 3^{-k} \right) = \sum_{k=1}^\infty (a_k/2) 2^{-k}.</math>

For any number y in [0,1], its binary representation can be translated into a ternary representation of a number x in C by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = ³/₅ = 0.100110011001...₂, we write x = 0.200220022002...₃ = ⁷/₁₀. Consequently f is surjective; however, f is not injective — interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, ⁷/₉ = 0.2022222...₃ and ⁸/₉ = 0.2200000...₃ so f(⁷/₉) = 0.101111...₂ = 0.11₂ = f(⁸/₉). So there are as many points in the Cantor set as there are in [0, 1], and the Cantor set is uncountable (see Cantor's diagonal argument). However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is ¼, which can be written as 0.02020202020...₃ in ternary notation. The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval. (Actually, the irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers, which are dense everywhere.)

Self-similarity

The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, there are two functions, the left and right self-similarity transformations, <math>f_L(x)=x/3</math> and <math>f_R(x)=(2+x)/3</math>, which leave the Cantor set invariant: <math>f_L(C)=f_R(C)=C</math>. Repeated iteration of <math>f_L</math> and <math>f_R</math> can be visualized as an infinite binary tree. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set <math>\{f_L,f_R\}</math> together with function composition forms a monoid, the dyadic monoid. The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, the Cantor set is a homogeneous space in the sense that for any two points <math>x</math> and <math>y</math> in the Cantor set <math>C</math>, there exists a homeomorphism <math>h:C\to C</math> with <math>h(x)=y</math>. These homeomorphisms can be expressed explicitly, as Mobius transformations. The Hausdorff dimension of the Cantor set is equal to <math>\ln(2)/\ln(3)</math>.

Topological and analytical properties

As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Since it is also totally bounded, the Heine-Borel theorem says that it must be compact. For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number with a ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point in the Cantor set is an accumulation point, but none is an interior point. A closed set in which every point is an accumulation point is also called a perfect set in topology, while a closed subset of the interval with no interior points is nowhere dense in the interval. Every point of the Cantor set is a cluster point of the Cantor set. Every point of the Cantor set is also a cluster point of the complement of the Cantor set. For two points in the Cantor set, there will be some ternary digit where they differ — one d will have 0 and the other 2. By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points. In the relative topology on the Cantor set, the points have been separated by a clopen set. Consequently the Cantor set is totally disconnected. As a compact totally disconnected Hausdorff space, the Cantor set is an example of a Stone space. As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of the space <math>\{0, 1\}</math>, where each copy carries the discrete topology. This is the space of all sequences in two digits: <math>2^\mathbb{N}=\{x_n\vert x_n\in \{0,1\}, n\in \mathbb{N}\}</math>, which can also be identified with the set of 2-adic integers. The basis for the open sets of the product topology are cylinder sets; the homeomorphism maps these to the subspace topology that the Cantor set inherits from the natural topology on the real number line. This characterization of the Cantor space as a product of compact spaces gives a second proof that Cantor space is compact, via Tychonoff's theorem. From the above characterization, the Cantor set is homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers. The Cantor set can be endowed with a metric, the p-adic metric. Given two sequences <math>\{x_n\},\{y_n\}\in 2^\mathbb{N}</math>, the distance between them may be given by <math>d(\{x_n\},\{y_n\}) = 1/k</math>, where <math>k</math> is the smallest index such that <math>x_k \ne y_k</math>; if there is no such index, then the two sequences are the same, and one defines the distance to be zero. This turns the Cantor set into a metric space. Every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set. See Cantor space for more on spaces homeomorphic to the Cantor set. The Cantor set is universal in the category of compact metric spaces. This means that any compact metric space is a continuous image of the Cantor set. This fact has important applications in functional analysis, where it is sometimes known as the representation theorem for compact metric spaces^[3].

Variants of the Cantor set

See main article Smith-Volterra-Cantor set.

Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. The resulting sets are all homeomorphic to the Cantor set and also have Lebesgue measure 0. In the case where the middle ⁸/₁₀ of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s. By removing progressively smaller percentages of the remaining pieces in every step, one can also construct sets homeomorphic to the Cantor set that have positive Lebesgue measure, while still being nowhere dense. See Smith-Volterra-Cantor set for an example.

Historical remarks

Cantor himself defined the set in a general, abstract way, and mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. The original paper provides several different constructions of the abstract concept. This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge. The discovery did much to set him on the course for developing an abstract, general theory of infinite sets.

See also

• Cantor function
• Cantor space
• Cantor cube
• Cantor dust

References

1. ^ Georg Cantor, On the Power of Perfect Sets of Points (De la puissance des ensembles parfait de points), Acta Mathematica 4 (1884) 381--392. English translation reprinted in Classics on Fractals, ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7
2. ^ Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos
3. ^ Stephen Willard, General Topology, Addison-Wesley Publishing Company, 1968.

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition). (See example 29).
• Gary L. Wise and Eric B. Hall, Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0-19-507068-2. (See chapter 1).
• Cantor Sets at cut-the-knot
• Cantor Set and Function at cut-the-knot