Inmathematics, theCantor set is aself-similar set of points lying on a singleline segment that has a number of unintuitive properties. It was discovered in 1874 byHenry John Stephen Smith[1][2][3][4] and mentioned by German mathematicianGeorg Cantor in 1883.[5][6] As it contrasts with alinear continuum, the Cantor set has been called theCantor discontinuum.[7]
Through consideration of this set, Cantor and others helped lay the foundations of modernpoint-set topology. The most common construction is theCantor 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 aperfect set that isnowhere dense.[5]
More generally, in topology, aCantor space is a topological spacehomeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturallyhomeomorphic to the countable product of thediscrete two-point space. By a theorem ofL. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact,metrizable and zero-dimensional.[8]
The Cantor ternary set is created by iteratively deleting theopen middle third from a set of line segments. One starts by deleting the open middle third from theinterval, 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 thisinfinite process. The same construction can be described recursively by setting
and
for, so that
The first six steps of this process are illustrated below.
Using the idea of self-similar transformations, and the explicit closed formulas for the Cantor set are[9]
where every middle third is removed as the open interval from theclosed 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 afinite subdivision rule. The complement of the Cantor ternary set is an example of afractal string.
In arithmetical terms, the Cantor set consists of allreal numbers of theunit interval that do not require the digit 1 in order to be expressed as aternary (base 3) fraction. Each point in the Cantor set is uniquely located by a path through an infinitely deepbinary 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.
InThe Fractal Geometry of Nature, mathematicianBenoit 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.[10]
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 thegeometric 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 removingopen sets (sets that do not include their endpoints). So removing the line segment from the original interval leaves behind the points1/3 and2/3. 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 notempty, and in fact contains anuncountably infinite number of points (as follows from the above description in terms of paths in an infinite binary tree).
It may appear thatonly the endpoints of the construction segments are left, but that is not the case either. The number1/4, for example, has the unique ternary form 0.020202... = 0.02. 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 of1/3.[11]All endpoints of segments areterminating ternary fractions and are contained in the set
which is acountably infinite set.As tocardinality,almost all elements of the Cantor set are not endpoints of intervals, norrational points like1/4. The whole Cantor set is in fact not countable.
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 isuncountable. To see this, we show that there is afunctionf from the Cantor set to the closed interval that issurjective (i.e.f maps from onto) so that the cardinality of is no less than that of. Since is asubset of, its cardinality is also no greater, so the two cardinalities must in fact be equal, by theCantor–Bernstein–Schröder theorem.
To construct this function, consider the points in the interval in terms of base 3 (orternary) notation. Recall that the proper ternary fractions, more precisely: the elements of, admit more than one representation in this notation, as for example1/3, that can be written as 0.13 = 0.103, but also as 0.0222...3 = 0.023, and2/3, that can be written as 0.23 = 0.203 but also as 0.1222...3 = 0.123.[12]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
This can be summarized by saying that those numbers with a ternary representation such that the first digit after theradix 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 firsttwo digits is 1.
Continuing in this way, for a number not to be excluded at stepn, it must have a ternary representation whosenth 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 1,1/3 = 0.13 and7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 0.23,1/3 = 0.0222...3 = 0.023 and7/9 = 0.20222...3 = 0.2023.All the latter numbers are "endpoints", and these examples are rightlimit points of. The same is true for the left limit points of, e.g.2/3 = 0.1222...3 = 0.123 = 0.203 and8/9 = 0.21222...3 = 0.2123 = 0.2203. All these endpoints areproper ternary fractions (elements of) of the formp/q, where denominatorq is apower of 3 when the fraction is in itsirreducible form.[11] 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 leftlimit 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 isdense in (but not dense in) and makes up acountably infinite set. The numbers in which arenot 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 abinary representation of a real number. In a formula,
For any numbery in, its binary representation can be translated into a ternary representation of a numberx in by replacing all the 1s by 2s. With this,f(x) =y so thaty is in therange off. For instance ify =3/5 = 0.100110011001...2 = 0.1001, we writex = 0.2002 = 0.200220022002...3 =7/10. Consequently,f is surjective. However,f isnotinjective — the values for whichf(x) coincides are those at opposing ends of one of themiddle thirds removed. For instance, take
so
Thus there are as many points in the Cantor set as there are in the interval (which has theuncountable 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 is1/4, which can be written as 0.020202...3 = 0.02 in ternary notation. In fact, given any, there exist such that. This was first demonstrated bySteinhaus in 1917, whoproved, via a geometric argument, the equivalent assertion that for every.[13] Since this construction provides an injection from to, we have as an immediatecorollary. Assuming that for any infinite set (a statement shown to be equivalent to theaxiom of choiceby Tarski), this provides another demonstration that.
The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. Theirrational numbers have the same property, but the Cantor set has the additional property of beingclosed, so it is not evendense in any interval, unlike the irrational numbers which are dense in every interval.
It has beenconjectured that allalgebraic irrational numbers arenormal. Since members of the Cantor set are not normal in base 3, this would imply that all members of the Cantor set are either rational ortranscendental.
The Cantor set is the prototype of afractal. It isself-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, and, which leave the Cantor set invariant up tohomeomorphism:
Repeatediteration of and can be visualized as an infinitebinary tree. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set together withfunction composition forms amonoid, thedyadic monoid.
Elements of the Cantor set can be associated with the2-adic integers, so as with usual integers, the automorphism group is themodular group. Thus theautomorphisms of the Cantor set arehyperbolic motions, particular isometries of thehyperbolic plane. Thus, the Cantor set is ahomogeneous space in the sense that for any two points and in the Cantor set, there exists a homeomorphism with. An explicit construction of can be described more easily if we see the Cantor setas a product space of countably many copies of the discrete space. Then the map defined by is aninvolutive homeomorphism exchanging and.
Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means anytopological space that ishomeomorphic (topologically equivalent) to it.
As the above summation argument shows, the Cantor set is uncountable but hasLebesgue measure 0. Since the Cantor set is thecomplement of aunion ofopen sets, it itself is aclosed subset of the reals, and therefore acomplete metric space. Since it is alsototally bounded, theHeine–Borel theorem says that it must becompact.
For any point in the Cantor set and any arbitrarily smallneighborhood 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 anaccumulation point (also called a cluster point or limit point) of the Cantor set, but none is aninterior point. A closed set in which every point is an accumulation point is also called aperfect set intopology, while a closed subset of the interval with no interior points isnowhere dense in the interval.
Every point of the Cantor set is also an accumulation point of the complement of the Cantor set.
For any two points in the Cantor set, there will be some ternary digit where they differ — one 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 therelative topology on the Cantor set, the points have been separated by aclopen set. Consequently, the Cantor set istotally disconnected. As a compact totally disconnectedHausdorff space, the Cantor set is an example of aStone space.
As a topological space, the Cantor set is naturallyhomeomorphic to theproduct of countably many copies of the space, where each copy carries thediscrete topology. This is the space of allsequences in two digits
which can also be identified with the set of2-adic integers. Thebasis for the open sets of theproduct topology arecylinder sets; the homeomorphism maps these to thesubspace topology that the Cantor set inherits from the natural topology on thereal line. This characterization of theCantor space as a product of compact spaces gives a second proof that Cantor space is compact, viaTychonoff's theorem.
From the above characterization, the Cantor set ishomeomorphic to thep-adic integers, and, if one point is removed from it, to thep-adic numbers.
The Cantor set is a subset of the reals, which are ametric space with respect to theordinary distance metric; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use thep-adic metric on: given two sequences, the distance between them is, where 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. These two metrics generate the sametopology on the Cantor set.
We have seen above that the Cantor set is a totally disconnectedperfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space ishomeomorphic to the Cantor set. SeeCantor space for more on spaceshomeomorphic to the Cantor set.
The Cantor set is sometimes regarded as "universal" in thecategory ofcompact metric spaces, since any compact metric space is acontinuousimage of the Cantor set; however this construction is not unique and so the Cantor set is notuniversal in the precisecategorical sense. The "universal" property has important applications infunctional analysis, where it is sometimes known as therepresentation theorem for compact metric spaces.[14]
For anyintegerq ≥ 2, the topology on thegroup G =Zqω (the countable direct sum) is discrete. Although thePontrjagin dual Γ is alsoZqω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it ishomeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the caseq = 2. (See Rudin 1962 p 40.)
The Cantor set can be seen as thecompact group of binary sequences, and as such, it is endowed with a naturalHaar measure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usualLebesgue measure on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of asingular measure. It can also be shown that the Haar measure is an image of anyprobability, making the Cantor set a universal probability space in some ways.
InLebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure.[15] In contrast, the set has aHausdorff measure of in its dimension of.[16]
If we define a Cantor number as a member of the Cantor set, then[17]
The Cantor set is ameagre set (or a set of first category) as a subset of (although not as a subset of itself, since it is aBaire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set, the Cantor set is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of. However, unlike, which is countable and has a "small" cardinality,, the cardinality of is the same as that of, the continuum, and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of that is meagre but of positive measure and a subset that is non-meagre but of measure zero:[18] By taking the countable union of "fat" Cantor sets of measure (see Smith–Volterra–Cantor set below for the construction), we obtain a setwhich has a positive measure (equal to 1) but is meagre in [0,1], since each is nowhere dense. Then consider the set. Since, cannot be meagre, but since, must have measure zero.
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. In the case where the middle8/10 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. If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder as for any such that.
On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration. Thus, one can construct setshomeomorphic to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length () is removed from the middle of each segment at thenth iteration, then the total length removed is, and the limiting set will have aLebesgue measure of. Thus, in a sense, the middle-thirds Cantor set is a limiting case with. If, then the remainder will have positive measure with. The case is known as theSmith–Volterra–Cantor set, which has a Lebesgue measure of.
Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finiteCartesian product of the Cantor set with itself, making it aCantor space. Like the Cantor set, Cantor dust haszero measure.[20]
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A different 2D analogue of the Cantor set is theSierpinski carpet, where a square is divided up into nine smaller squares, and the middle one removed. The remaining squares are then further divided into nine each and the middle removed, and so on ad infinitum.[21] One 3D analogue of this is theMenger sponge.
Cantor introduced what we call today the Cantor ternary set as an example "of aperfect point-set, which is not everywhere-dense in any interval, however small."[22][23] Cantor described in terms of ternary expansions, as "the set of all real numbers given by the formula:where the coefficients arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."[22]
A topological space is perfect if all its points are limit points or, equivalently, if it coincides with itsderived set. Subsets of the real line, like, can be seen as topological spaces under the induced subspace topology.[8]
Cantor was led to the study of derived sets by his results on uniqueness oftrigonometric series.[23] The latter did much to set him on the course for developing anabstract, general theory of infinite sets.
Benoit Mandelbrot wrote much on Cantor dusts and their relation tonatural fractals andstatistical physics.[10] He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. InThe Fractal Geometry of Nature, he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as theKoch andPeano curves," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming to be interesting in science."[10]
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