In mathematics, the Cantor set is a self-similar set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen SmithThe "Cantor set" was also discovered by Paul du Bois-Reymond (1831–1889). See . The "Cantor set" was also discovered in 1881 by Vito Volterra (1860–1940). See: . and mentioned by German mathematician Georg Cantor in 1883. As it contrasts with a linear continuum, the Cantor set has been called the Cantor discontinuum.Kazimierz Kuratowski (1972) Leo F. Boron, translator, Introduction to Set Theory and Topology, second edition, ch XVI, § 8 The Cantor Discontinuum, page 210 to 15, Pergamon Press
Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense.
More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product of the discrete two-point space . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional.
Construction and formula of the ternary set The Cantor ternary set is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third from the interval , leaving two line segments: . Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: . The Cantor ternary set contains all points in the interval that are not deleted at any step in this infinite process. The same construction can be described recursively by setting
and
for , so that for any .
The first six steps of this process are illustrated below.
The explicit closed formulas for the Cantor set are
where every middle third is removed as the open interval from the closed interval surrounding it, or
where the middle third of the foregoing closed interval is removed by intersecting with
This process of removing middle thirds is a simple example of a finite subdivision rule. The complement of the Cantor ternary set is an example of a fractal string.
In arithmetical terms, the Cantor set consists of all real numbers of the unit interval that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction. Each point in the Cantor set is uniquely located by a path through an infinitely deep binary tree, where the path turns left or right at each level according to which side of a deleted segment the point lies on. Representing each left turn with 0 and each right turn with 2 yields the ternary fraction for a point. Requiring the digit 1 is critical: , which is included in the Cantor set, can be written as , but also as , which contains no 1 digits and corresponds to an initial left turn followed by infinitely many right turns in the binary tree.
Mandelbrot's construction by "curdling" In The Fractal Geometry of Nature, mathematician Benoit Mandelbrot provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of . His narrative begins with imagining a bar, perhaps of lightweight metal, in which the bar's matter "curdles" by iteratively shifting towards its extremities. As the bar's segments become smaller, they become thin, dense slugs that eventually grow too small and faint to see.CURDLING: The construction of the Cantor bar results from the process I call curdling. It begins with a round bar. It is best to think of it as having a very low density. Then matter "curdles" out of this bar's middle third into the end thirds, so that the positions of the latter remain unchanged. Next matter curdles out of the middle third of each end third into its end thirds, and so on ad infinitum until one is left with an infinitely large number of infinitely thin slugs of infinitely high density. These slugs are spaced along the line in the very specific fashion induced by the generating process. In this illustration, curdling (which eventually requires hammering!) stops when both the printer's press and our eye cease to follow; the last line is indistinguishable from the last but one: each of its ultimate parts is seen as a gray slug rather than two parallel black slugs.
Composition Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit interval remaining can be found by total length removed. This total is the geometric progression
So that the proportion left is .
This calculation suggests that the Cantor set cannot contain any interval of non-zero length. 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 from the original interval leaves behind the points and . 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 uncountably infinite number of points (as follows from the above description in terms of paths in an infinite binary tree).
It may appear that only the endpoints of the construction segments are left, but that is not the case either. The number , for example, has the unique ternary form 0.020202... = . It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of . All endpoints of segments are terminating ternary fractions and are contained in the set
which is a countably infinite set. As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, nor rational points like . The whole Cantor set is in fact not countable.
Properties
Cardinality It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set to the closed interval that is surjective (i.e. f maps from onto ) so that the cardinality of is no less than that of . Since is a subset of , its cardinality is also no greater, so the two cardinalities must in fact be equal, by the Cantor–Bernstein–Schröder theorem.
To construct this function, consider the points in the interval in terms of base 3 (or ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of , admit more than one representation in this notation, as for example , that can be written as 0.13 = 3, but also as 0.0222...3 = 3, and , that can be written as 0.23 = 3 but also as 0.1222...3 = 3.This alternative recurring representation of a number with a terminating numeral occurs in any positional system with Archimedean absolute value. When we remove the middle third, this 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 consist of Numbers of the form 0.0xxxxx...3 (including 0.022222...3 = 1/3) Numbers of the form 0.2xxxxx...3 (including 0.222222...3 = 1)
This can be summarized by saying that those numbers with a ternary representation such that the first digit after the radix point is not 1 are the ones remaining after the first step.
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 where neither of the first two digits is 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 admit a numeral representation consisting entirely of 0s and 2s.
It is worth emphasizing that numbers like 1, = 0.13 and = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 3, = 0.0222...3 = 3 and = 0.20222...3 = 3. All the latter numbers are "endpoints", and these examples are right limit points of . The same is true for the left limit points of , e.g. = 0.1222...3 = 3 = 3 and = 0.21222...3 = 3 = 3. All these endpoints are proper ternary fractions (elements of ) of the form , where denominator q is a power of 3 when the fraction is in its irreducible form. The ternary representation of these fractions terminates (i.e., is finite) or — recall from above that proper ternary fractions each have 2 representations — is infinite and "ends" in either infinitely many recurring 0s or infinitely many recurring 2s. Such a fraction is a left limit point of if its ternary representation contains no 1's and "ends" in infinitely many recurring 0s. Similarly, a proper ternary fraction is a right limit point of if it again its ternary expansion contains no 1's and "ends" in infinitely many recurring 2s.
This set of endpoints is dense in (but not dense in ) and makes up a countably infinite set. The numbers in which are not endpoints also have only 0s and 2s in their ternary representation, but they cannot end in an infinite repetition of the digit 0, nor of the digit 2, because then it would be an endpoint.
The function from to is defined by taking the ternary numerals that do 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,
where
For any number y in , its binary representation can be translated into a ternary representation of a number x in 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 = = 0.100110011001...2 = , we write x = = 0.200220022002...3 = . Consequently, f is surjective. However, f is not injective — the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, take = 3 (which is a right limit point of and a left limit point of the middle third [, ]) and = 3 (which is a left limit point of and a right limit point of the middle third [, ]) so
Thus there are as many points in the Cantor set as there are in the interval (which has the uncountable cardinality 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 , which can be written as 0.020202...3 = in ternary notation. In fact, given any , there exist such that . This was first demonstrated by Steinhaus in 1917, who proved, via a geometric argument, the equivalent assertion that for every . Since this construction provides an injection from to , we have as an immediate corollary. Assuming that for any infinite set (a statement shown to be equivalent to the axiom of choice by Tarski), this provides another demonstration that .