Cantor set

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In mathematics, the Cantor set, introduced by German mathematician Georg Cantor in 1883^[1], 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

[edit] 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 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. 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.

[edit] 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

So that the proportion left is 1 – 1 = 0. Alternatively, it can be observed that each step leaves 2/3 of the length in the previous stage, so that the amount remaining is 2/3 × 2/3 × 2/3 × ..., an infinite product with a limit of 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.

[edit] Properties

[edit] 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 0's and 2's. 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 0's and 2's: 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, only one of its numerals consists entirely of 0's and 2's.

The function from C to [0,1] is defined by taking the numeral that does consist entirely of 0's and 2's, and replacing all the 2's by 1's. In a formula,

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 1's by 2's. 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).

In other words, the "endpoints" of the Cantor set are all numbers with ternary representation consisting of only 0's and 2's. Since there is a clear bijection between the ternary numbers consisting of only the digits 0 and 2 and the binary numbers consisting of the digits 0 and 1, it follows that the number of endpoints in the Cantor set is equal to the number of binary strings. The number of binary strings is uncountable by Cantor's diagonal argument, thus the Cantor set contains an uncountable number of points, though it contains no interval. This is the "paradox" of the Cantor set, that it contains as many points as the interval from which it is taken, yet it 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.)

[edit] 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, f_L(x) = x / 3 and f_R(x) = (2 + x) / 3, which leave the Cantor set invariant: f_L(C) = f_R(C) = C.

Repeated iteration of f_L and f_R 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 {f_L,f_R} 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 x and y in the Cantor set C, there exists a homeomorphism with h(x) = y. These homeomorphisms have explicit form as Mobius transformations.

Its Hausdorff dimension is equal to ln(2) / ln(3).

[edit] 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 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 0's and 2's, as well as numbers whose ternary numerals contain 1's. 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 {0,1}, where each copy carries the discrete topology. This is the space of all sequences in two digits: . This is, of course, 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.

From the above characterization, the Cantor set homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers.

The Cantor set is compact; this is a consequence of Tychonoff's theorem, as it is a product of (a countable number of copies of) the compact set {0,1}.

The Cantor set can be endowed with a metric, the p-adic metric. Given two sequences , the distance between them may be given by d({x_n},{y_n}) = 1 / k, where k is the smallest index such that ; 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^[2].

[edit] 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 0's and 9's.

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.

[edit] 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.

[edit] See also

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

[edit] 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. ^ 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