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OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Cantor set is a set of points lying on a single
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern
point-set topology In mathematics, general 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 differential topology, geometr ...
. 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 the ternary construction only in passing, as an example of a more general idea, that of a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. In ...
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 " ...
is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of Brouwer, this is equivalent to being perfect nonempty, compact metrizable 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 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
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 analysis ...
binary tree In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
, 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.

# 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 :$\sum_^\infty \frac = \frac + \frac + \frac + \frac + \cdots = \frac\left\left(\frac\right\right) = 1.$ 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 sets (sets that do not include their endpoints). So removing the line segment (, ) from the original interval , 1leaves 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 1/3. All endpoints of segments are ''terminating'' ternary fractions and are contained in the set :$\left\ \qquad \Bigl\left(\subset \N_0 \, 3^ \Bigr\right)$ 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 number ...
set. As to
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
,
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathe ...
elements of the Cantor set are not endpoints of intervals, nor rational points like 1/4. 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 (or uncountably infinite set) 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 nu ...
. To see this, we show that there is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
''f'' from the Cantor set $\mathcal$ to the closed interval ,1that is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
(i.e. ''f'' maps from $\mathcal$ onto ,1 so that the cardinality of $\mathcal$ is no less than that of ,1 Since $\mathcal$ is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...
of ,1 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 , 1interval in terms of base 3 (or ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of $\bigl\left(\Z \setminus \\bigr\right) \cdot 3^$, 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 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 with respect to the topology on X also con ...
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 with respect to the topology on X also con ...
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 substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in $\mathcal$ (but not dense in , 1 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 number ...
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 ,1is 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, :$f \bigg\left( \sum_ a_k 3^ \bigg\right) = \sum_ \frac 2^$   where   $\forall k\in \N : a_k \in \ .$ For any number ''y'' in ,1 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 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; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
— 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 :$\begin f\bigl\left(^1\!\!/\!_3 \bigr\right) = f\left(0.0\overline_3\right) = 0.0\overline_2 = \!\! & \!\! 0.1_2 \!\! & \!\! = 0.1\overline_2 = f\left(0.2\overline_3\right) = f\bigl\left(^2\!\!/\!_3 \bigr\right) . \\ & \parallel \\ & ^1\!\!/\!_2 \end$ Thus there are as many points in the Cantor set as there are in the interval , 1(which has the
uncountable In mathematics, an uncountable set (or uncountably infinite set) 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 nu ...
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
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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...
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 (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
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 substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in any interval, unlike the irrational numbers which are dense in every interval. It has been
conjecture In mathematics, a conjecture is a Consequent, conclusion or a proposition that is proffered on a tentative basis without Formal proof, proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conje ...
d that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal, 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 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 scales, as illus ...
. It is
self-similar __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically s ...
, 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, $T_L\left(x\right)=x/3$ and $T_R\left(x\right)=\left(2+x\right)/3$, which leave the Cantor set invariant up to
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
: $T_L\left(\mathcal\right)\cong T_R\left(\mathcal\right)\cong \mathcal=T_L\left(\mathcal\right)\cup T_R\left(\mathcal\right).$ 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 k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary t ...
. 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, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
forms a
monoid In abstract algebra, a branch of mathematics, 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 0. Monoids a ...
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 of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
. Thus, the Cantor set is a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
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 $h\left(x\right)=y$. 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 $h_n\left(u\right):=u_n+x_n+y_n \mod 2$ is an involutive homeomorphism exchanging $x$ and $y$.

## Conservation law

It has been found that some form of conservation law is always responsible behind scaling and self-similarity. In the case of Cantor set it can be seen that the $d_f$th moment (where $d_f=\ln\left(2\right)/\ln\left(3\right)$ is the
fractal dimension In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
) of all the surviving intervals at any stage of the construction process is equal to constant which is equal to one in the case of Cantor set. We know that there are $N=2^n$ intervals of size $1/3^n$ present in the system at the $n$th step of its construction. Then if we label the surviving intervals as $x_1, x_2, \ldots, x_$ then the $d_f$th moment is $x_1^+x_2^+\cdots+x_^=1$ since $x_1=x_2= \cdots =x_=1/3^n$. The
Hausdorff dimension In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
of the Cantor set is equal to ln(2)/ln(3) ≈ 0.631.

## 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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
that is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
(topologically equivalent) to it. As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since the Cantor set is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact. For any point in the Cantor set and any arbitrarily small
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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 (also called a cluster point or limit point) of the Cantor set, but none is an interior point. A closed set in which every point is an accumulation point is also called a
perfect set In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all limit points of S, also known as the derived set of S. In ...
in
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 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 counter-intuitive, as the common meanings of and are antonyms, but their mathematical ...
. Consequently, the Cantor set is totally disconnected. As a compact totally disconnected
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
, 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 $\$, 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 call ...
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-seemi ...
are
cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod ...
s; the homeomorphism maps these to the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
that the Cantor set inherits from the natural topology on the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. 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 " ...
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 tran ...
. From the above characterization, the Cantor set is homeomorphic to the ''p''-adic integers, and, if one point is removed from it, to the ''p''-adic numbers. The Cantor set is a subset of the reals, which are a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
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 $\left(x_n\right),\left(y_n\right)\in 2^\mathbb$, the distance between them is $d\left(\left(x_n\right),\left(y_n\right)\right) = 2^$, 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 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 " ...
for more on spaces homeomorphic to the Cantor set. The Cantor set is sometimes regarded as "universal" in the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
of compact metric spaces, since any compact metric space is a continuous
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, where it is sometimes known as the ''representation theorem for compact metric spaces''. For any integer ''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 ide ...
G = Z''q''ω (the countable direct sum) is discrete. Although the Pontrjagin dual 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 to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case ''q'' = 2. (See Rudin 1962 p 40.) The
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of the Cantor set is approximately 0.274974.

## 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 was introduced by Alfréd Haar in 1933, thou ...
. 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 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. It can also be shown that the Haar measure is an image of any
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
, making the Cantor set a universal probability space in some ways. In Lebesgue measure 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 of 1 in its dimension of log 2 / log 3.

## Cantor numbers

If we define a Cantor number as a member of the Cantor set, then # Every real number in , 2is 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 field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called ...
(or a set of first category) as a subset of ,1(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

# 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 $\left(1-f\right)^n\to 0$ as $n\to\infty$ for any $f$ such that
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-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–Vo ...
, which has a Lebesgue measure of $1/2$.

## Stochastic Cantor set

One can modify the construction of the Cantor set by dividing randomly instead of equally. Besides, to incorporate time we can divide only one of the available intervals at each step instead of dividing all the available intervals. In the case of stochastic triadic Cantor set the resulting process can be described by the following rate equation :$\frac =-\frac c\left(x,t\right) + 2\int_x^\infty \left(y-x\right)c\left(y,t\right) \, dy,$ and for the stochastic dyadic Cantor set :$=-xc\left(x,t\right)+\left(1+p\right)\int_x^\infty c\left(y,t\right) \, dy,$ where $c\left(x,t\right)dx$ is the number of intervals of size between $x$ and $x+dx$. In the case of triadic Cantor set the fractal dimension is $0.5616$ which is less than its deterministic counterpart $0.6309$. In the case of stochastic dyadic Cantor set the fractal dimension is $p$ which is again less than that of its deterministic counterpart $\ln \left(1+p\right)/\ln 2$. In the case of stochastic dyadic Cantor set the solution for $c\left(x,t\right)$ exhibits dynamic scaling as its solution in the long-time limit is $t^e^$ where the fractal dimension of the stochastic dyadic Cantor set $d_f=p$. In either case, like triadic Cantor set, the $d_f$th moment ($\int x^ c(x,t) \, dx = \text$) of stochastic triadic and dyadic Cantor set too are conserved quantities.

## 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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
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 " ...
. 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 Si ...
.

# 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 $P\text{'}$. 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. The latter did much to set him on the course for developing an abstract, general theory of infinite sets.

* The indicator function of the Cantor set *
Smith–Volterra–Cantor set In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-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–Vo ...
* Cantor function * 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 According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illu ...
* Moser–de Bruijn sequence * * * * * . *