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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Cantor set is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of points lying on a single
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern
point-set topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
. 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 In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
is a topological space
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the countable product ^ of the discrete two-point space \underline 2 . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact,
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
and zero-dimensional.


Construction and formula of the ternary set

The Cantor ternary set \mathcal is created by iteratively deleting the ''open'' middle third from a set of line segments. One starts by deleting the open middle third \left(\frac, \frac\right) from the interval \textstyle\left , 1\right/math>, leaving two line segments: \left , \frac\rightcup\left frac, 1\right/math>. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: \left , \frac\rightcup\left frac, \frac\rightcup\left frac, \frac\rightcup\left frac, 1\right/math>. The Cantor ternary set contains all points in the interval ,1/math> that are not deleted at any step in this infinite process. The same construction can be described recursively by setting : C_0 := ,1/math> and : C_n := \frac 3 \cup \left(\frac 2 +\frac 3 \right) = \frac13 \bigl(C_ \cup \left(2 + C_ \right)\bigr) for n \ge 1, so that : \mathcal := = \bigcap_^\infty C_n = \bigcap_^\infty C_n   for any   m \ge 0. The first six steps of this process are illustrated below. Using the idea of self-similar transformations, T_L(x)=x/3, T_R(x)=(2+x)/3 and C_n =T_L(C_)\cup T_R(C_), the explicit closed formulas for the Cantor set are : \mathcal= ,1\,\setminus\, \bigcup_^\infty \bigcup_^ \left(\frac,\frac \right)\!, where every middle third is removed as the open interval \left(\frac,\frac\right) from the
closed interval In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
\left frac,\frac\right= \left frac,\frac\right/math> surrounding it, or : \mathcal=\bigcap_^\infty \bigcup_^ \left( \left frac,\frac\right\cup \left frac,\frac\right\right)\!, where the middle third \left(\frac,\frac\right) of the foregoing closed interval \left frac,\frac\right= \left frac,\frac\right/math> is removed by intersecting with \left frac,\frac\right\cup \left frac,\frac\right!. This process of removing middle thirds is a simple example of a
finite subdivision rule In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeati ...
. The complement of the Cantor ternary set is an example of a fractal string. In arithmetical terms, the Cantor set consists of all
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s of the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
,1/math> that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction. As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep
binary tree In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...
, 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: \frac, which is included in the Cantor set, can be written as 0.1, but also as 0.0\bar, 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 ''The Fractal Geometry of Nature'' is a 1982 book by the Franco-American mathematician Benoît Mandelbrot. Overview ''The Fractal Geometry of Nature'' is a revised and enlarged version of his 1977 book entitled ''Fractals: Form, Chance and Dime ...
'', mathematician
Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of \mathcal. 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 A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the s ...
:\sum_^\infty \frac = \frac + \frac + \frac + \frac + \cdots = \frac\left(\frac\right) = 1. So that the proportion left is 1 - 1 = 0. 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 set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s (sets that do not include their endpoints). So removing the line segment \left(\frac, \frac\right) from the original interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> 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 In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
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 : \left\ \qquad \Bigl(\subset \N_0 \, 3^ \Bigr) which is a
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
set. As to
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
,
almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
elements of the Cantor set are not endpoints of intervals, nor
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
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 In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
. To see this, we show that there is a function ''f'' from the Cantor set \mathcal to the closed interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> that is
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
(i.e. ''f'' maps from \mathcal onto
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>) so that the cardinality of \mathcal is no less than that of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>. Since \mathcal is a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, 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
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> interval in terms of base 3 (or ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of \bigl(\Z \setminus \\bigr) \cdot 3^, 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. 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 alt=Four types of separating decimals: a) 1,234.56. b) 1.234,56. c) 1'234,56. d) ١٬٢٣٤٫٥٦., Both a full_stop.html" ;"title="comma and a full stop">comma and a full stop (or period) are generally accepted decimal separators for interna ...
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 ''n''th 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 point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
s of \mathcal. The same is true for the left limit points of \mathcal, e.g. = 0.1222...3 = 3 = 3 and = 0.21222...3 = 3 = 3. All these endpoints are ''proper ternary'' fractions (elements of \Z \cdot 3^) 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 In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x contains a point of S other than x itself. A ...
of \mathcal 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 \mathcal if it again its ternary expansion contains no 1's and "ends" in infinitely many recurring 2s. This set of endpoints is
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in \mathcal (but not dense in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>) and makes up a
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
set. The numbers in \mathcal 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 \mathcal to
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> 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, :f \bigg( \sum_ a_k 3^ \bigg) = \sum_ \frac 2^   where   \forall k\in \N : a_k \in \ . For any number ''y'' in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, its binary representation can be translated into a ternary representation of a number ''x'' in \mathcal by replacing all the 1s by 2s. With this, ''f''(''x'') = ''y'' so that ''y'' is in the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of ''f''. For instance if ''y'' = = 0.100110011001...2 = , we write ''x'' = = 0.200220022002...3 = . Consequently, ''f'' is surjective. However, ''f'' is ''not''
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
— 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 \mathcal and a left limit point of the middle third   and : = 3 (which is a left limit point of \mathcal and a right limit point of the middle third so :\begin f\bigl(^1\!\!/\!_3 \bigr) = f(0.0\overline_3) = 0.0\overline_2 = \!\! & \!\! 0.1_2 \!\! & \!\! = 0.1\overline_2 = f(0.2\overline_3) = f\bigl(^2\!\!/\!_3 \bigr) . \\ & \parallel \\ & ^1\!\!/\!_2 \end Thus there are as many points in the Cantor set as there are in the interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> (which has the
uncountable In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger tha ...
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 a\in 1,1/math>, there exist x,y\in\mathcal such that a = y-x. This was first demonstrated by Steinhaus in 1917, who proved, via a geometric argument, the equivalent assertion that \ \; \cap \; (\mathcal\times\mathcal) \neq\emptyset for every a\in 1,1/math>. Since this construction provides an injection from 1,1/math> to \mathcal\times\mathcal, we have , \mathcal\times\mathcal, \geq, 1,1=\mathfrak as an immediate
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
. Assuming that , A\times A, =, A, for any infinite set A (a statement shown to be equivalent to the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
by Tarski), this provides another demonstration that , \mathcal, =\mathfrak. The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
s have the same property, but the Cantor set has the additional property of being closed, so it is not even
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in any interval, unlike the irrational numbers which are dense in every interval. It has been
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
d that all algebraic irrational numbers are normal. 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 or transcendental.


Self-similarity

The Cantor set is the prototype of a
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
. 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, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, T_L(x)=x/3 and T_R(x)=(2+x)/3, which leave the Cantor set invariant up to
homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
: T_L(\mathcal)\cong T_R(\mathcal)\cong \mathcal=T_L(\mathcal)\cup T_R(\mathcal). Repeated
iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
of T_L and T_R can be visualized as an infinite
binary tree In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theor ...
. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set \ together with
function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...
forms a
monoid In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity ...
, the
dyadic monoid In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on th ...
. The
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s of the binary tree are its hyperbolic rotations, and are given by the
modular group In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
. Thus, the Cantor set is a
homogeneous space In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
in the sense that for any two points x and y in the Cantor set \mathcal, there exists a homeomorphism h:\mathcal\to \mathcal with h(x)=y. An explicit construction of h can be described more easily if we see the Cantor set as a product space of countably many copies of the discrete space \. Then the map h:\^\N\to\^\N defined by h_n(u):=u_n+x_n+y_n \mod 2 is an involutive homeomorphism exchanging x and y.


Topological and analytical properties

Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means any
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
that is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
(topologically equivalent) to it. As the above summation argument shows, the Cantor set is uncountable but has
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
0. Since the Cantor set is the complement of a union of
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s, it itself is a closed subset of the reals, and therefore a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
. Since it is also
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...
, the
Heine–Borel theorem In real analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space \mathbb^n, the following two statements are equivalent: *S is compact, that is, every open cover of S has a finite s ...
says that it must be
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
. For any point in the Cantor set and any arbitrarily small
neighborhood A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
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 In mathematics, a limit point, accumulation point, or cluster point of a Set (mathematics), set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood (mathematics), neighbourhood of ...
(also called a cluster point or limit point) of the Cantor set, but none is an
interior point In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of t ...
. A closed set in which every point is an accumulation point is also called a perfect set in
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, while a closed subset of the interval with no interior points is nowhere 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 the
relative topology Relative may refer to: General use *Kinship and family, the principle binding the most basic social units of society. If two people are connected by circumstances of birth, they are said to be ''relatives''. Philosophy * Relativism, the concept ...
on the Cantor set, the points have been separated by a
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical de ...
. Consequently, the Cantor set is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
. As a compact totally disconnected
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
, the Cantor set is an example of a Stone space. As a topological space, the Cantor set is naturally
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the product of countably many copies of the space \, where each copy carries the
discrete topology In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
. This is the space of all
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s in two digits :2^\mathbb = \, which can also be identified with the set of 2-adic integers. The basis for the open sets of the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
are cylinder sets; the homeomorphism maps these to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
that the Cantor set inherits from the natural topology on the
real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
. This characterization of the
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
as a product of compact spaces gives a second proof that Cantor space is compact, via
Tychonoff's theorem In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is tra ...
. From the above characterization, the Cantor set is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the ''p''-adic integers, and, if one point is removed from it, to the ''p''-adic numbers. The Cantor set is a subset of the reals, which are a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
with respect to the ordinary distance metric; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the ''p''-adic metric on 2^\mathbb: given two sequences (x_n),(y_n)\in 2^\mathbb, the distance between them is d((x_n),(y_n)) = 2^, where k is the smallest index such that x_k \ne y_k; 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 same
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on the Cantor set. We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the Cantor set. See
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
for more on spaces
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the Cantor set. The Cantor set is sometimes regarded as "universal" in the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
of
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
metric spaces, since any compact metric space is a continuous
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of the Cantor set; however this construction is not unique and so the Cantor set is not universal in the precise categorical sense. The "universal" property has important applications in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, where it is sometimes known as the ''representation theorem for compact metric spaces''. For any
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''q'' ≥ 2, the topology on the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
G = Z''q''ω (the countable direct sum) is discrete. Although the
Pontrjagin dual In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one ...
Γ is also Z''q''ω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case ''q'' = 2. (See Rudin 1962 p 40.)


Measure and probability

The Cantor set can be seen as the
compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
of binary sequences, and as such, it is endowed with a natural
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfr� ...
. 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 usual
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
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 a
singular measure In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero o ...
. It can also be shown that the Haar measure is an image of any
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
, making the Cantor set a universal probability space in some ways. In
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
theory, the Cantor set is an example of a set which is uncountable and has zero measure. In contrast, the set has a
Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assi ...
of 1 in its dimension of \log_3(2).


Cantor numbers

If we define a Cantor number as a member of the Cantor set, then # Every real number in , 2/math> is the sum of two Cantor numbers. # Between any two Cantor numbers there is a number that is not a Cantor number.


Descriptive set theory

The Cantor set is a
meagre set In the mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itse ...
(or a set of first category) as a subset of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> (although not as a subset of itself, since it is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set \mathbb\cap ,1/math>, the Cantor set \mathcal is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>. However, unlike \mathbb\cap ,1/math>, which is countable and has a "small" cardinality, \aleph_0, the cardinality of \mathcal is the same as that of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, the continuum \mathfrak, and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> that is meagre but of positive measure and a subset that is non-meagre but of measure zero: By taking the countable union of "fat" Cantor sets \mathcal^ of measure \lambda = (n-1)/n (see Smith–Volterra–Cantor set below for the construction), we obtain a set \mathcal := \bigcup_^\mathcal^which has a positive measure (equal to 1) but is meagre in ,1 since each \mathcal^ is nowhere dense. Then consider the set \mathcal^ = ,1\setminus\bigcup_^\infty \mathcal^. Since \mathcal\cup\mathcal^ = ,1/math>, \mathcal^ cannot be meagre, but since \mu(\mathcal)=1, \mathcal^ must have measure zero.


Variants


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. In the case where the middle of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in ,1that 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 (1-f)^n\to 0 as n\to\infty for any f such that 0. 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 sets
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length r^n (r\leq 1/3) is removed from the middle of each segment at the ''n''th iteration, then the total length removed is \sum_^\infty 2^r^n=r/(1-2r), and the limiting set will have a
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
of \lambda=(1-3r)/(1-2r). Thus, in a sense, the middle-thirds Cantor set is a limiting case with r=1/3. If 0, then the remainder will have positive measure with 0<\lambda<1. The case r=1/4 is known as the
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
, which has a Lebesgue measure of 1/2.


Cantor dust

Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finite
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
of the Cantor set with itself, making it a
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
. Like the Cantor set, Cantor dust has zero measure. A different 2D analogue of the Cantor set is the Sierpinski 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. One 3D analogue of this is the
Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sie ...
.


Historical remarks

Cantor introduced what we call today the Cantor ternary set \mathcal C as an example "of a perfect point-set, which is not everywhere-dense in any interval, however small." Cantor described \mathcal C in terms of ternary expansions, as "the set of all real numbers given by the formula: z=c_1/3 +c_2/3^2 + \cdots + c_\nu/3^\nu +\cdots where the coefficients c_\nu arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements." A topological space P is perfect if all its points are limit points or, equivalently, if it coincides with its derived set P'. Subsets of the real line, like \mathcal C, can be seen as topological spaces under the induced subspace topology. Cantor was led to the study of derived sets by his results on uniqueness of
trigonometric series In mathematics, trigonometric series are a special class of orthogonal series of the form : A_0 + \sum_^\infty A_n \cos + B_n \sin, where x is the variable and \ and \ are coefficients. It is an infinite version of a trigonometric polynom ...
. The latter did much to set him on the course for developing an abstract, general theory of infinite sets.
Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...
wrote much on Cantor dusts and their relation to natural fractals and
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. In
The Fractal Geometry of Nature ''The Fractal Geometry of Nature'' is a 1982 book by the Franco-American mathematician Benoît Mandelbrot. Overview ''The Fractal Geometry of Nature'' is a revised and enlarged version of his 1977 book entitled ''Fractals: Form, Chance and Dime ...
, he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the Koch and Peano 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 \mathcal C to be interesting in science."


See also

* The indicator function of the Cantor set *
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–V ...
*
Cantor function In mathematics, the Cantor function is an example of a function (mathematics), function that is continuous function, continuous, but not absolute continuity, absolutely continuous. It is a notorious Pathological_(mathematics)#Pathological_exampl ...
* Cantor cube * Antoine's necklace *
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Cur ...
* Knaster–Kuratowski fan * List of fractals by Hausdorff dimension * Moser–de Bruijn sequence


Notes


References

* * * * * . *


External links

*
Cantor Sets
an
Cantor Set and Function
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

Cantor Set
at Platonic Realms {{DEFAULTSORT:Cantor Set Measure theory Topological spaces Sets of real numbers Georg Cantor L-systems