In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician

_{3} = _{3}, but also as 0.0222..._{3} = _{3}, and , that can be written as 0.2_{3} = _{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 _{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, = 0.1_{3} and = 0.21_{3} 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 _{3} = _{3} = _{3} and = 0.21222..._{3} = _{3} = _{3}. All these endpoints are ''proper ternary'' fractions (elements of $\backslash Z\; \backslash cdot\; 3^$) 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 _{2} = , we write ''x'' = = 0.200220022002..._{3} = . Consequently, ''f'' is surjective. However, ''f'' is ''not'' injective — the values for which ''f''(''x'') coincides are those at opposing ends of one of the ''middle thirds'' removed. For instance, take
: = _{3} (which is a right limit point of $\backslash mathcal$ and a left limit point of the middle third and
: = _{3} (which is a left limit point of $\backslash mathcal$ and a right limit point of the middle third
so
:$\backslash begin\; f\backslash bigl(^1\backslash !\backslash !/\backslash !\_3\; \backslash bigr)\; =\; f(0.0\backslash overline\_3)\; =\; 0.0\backslash overline\_2\; =\; \backslash !\backslash !\; \&\; \backslash !\backslash !\; 0.1\_2\; \backslash !\backslash !\; \&\; \backslash !\backslash !\; =\; 0.1\backslash overline\_2\; =\; f(0.2\backslash overline\_3)\; =\; f\backslash bigl(^2\backslash !\backslash !/\backslash !\_3\; \backslash bigr)\; .\; \backslash \backslash \; \&\; \backslash parallel\; \backslash \backslash \; \&\; ^1\backslash !\backslash !/\backslash !\_2\; \backslash end$
Thus there are as many points in the Cantor set as there are in the interval , 1(which has the _{3} = in ternary notation. In fact, given any $a\backslash in;\; href="/html/ALL/l/1,1.html"\; ;"title="1,1">1,1$

_{''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 of the Cantor set is approximately 0.274974.

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

Cantor Sets

an

Cantor Set and Function

at cut-the-knot

Cantor Set

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

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 of ...

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, geomet ...

. 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 point, limit points of S, also known as the derived set ...

that is nowhere dense
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...

.
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 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 $\backslash mathcal$ is created by iteratively deleting the ''open'' middle third from a set of line segments. One starts by deleting the open middle third $\backslash left(\backslash frac,\; \backslash frac\backslash right)$ from the interval $\backslash textstyle\backslash left;\; href="/html/ALL/l/,\_1\backslash right.html"\; ;"title=",\; 1\backslash right">,\; 1\backslash right$closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...

$\backslash left;\; href="/html/ALL/l/frac,\backslash frac\backslash right.html"\; ;"title="frac,\backslash frac\backslash right">frac,\backslash frac\backslash right$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
An ordinary fractal string \Omega is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths \mathcal=\ written in non-increasing order; we ...

.
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. Every ...

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 ...

$;\; href="/html/ALL/l/,1.html"\; ;"title=",1">,1$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 thegeometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...

:$\backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots\; =\; \backslash frac\backslash left(\backslash frac\backslash right)\; =\; 1.$
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, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

s (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
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...

, and in fact contains an uncountably infinite
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 num ...

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
:$\backslash left\backslash \; \backslash qquad\; \backslash Bigl(\backslash subset\; \backslash N\_0\; \backslash ,\; 3^\; \backslash Bigr)$
which is a countably infinite set.
As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, nor rational
Rationality is the quality of being guided by or based on reasons. 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 abi ...

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 isuncountable
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 num ...

. 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-oriente ...

''f'' from the Cantor set $\backslash mathcal$ to the closed interval ,1that is surjective (i.e. ''f'' maps from $\backslash mathcal$ onto ,1 so that the cardinality of $\backslash mathcal$ is no less than that of ,1 Since $\backslash mathcal$ is a subset 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 $\backslash bigl(\backslash Z\; \backslash setminus\; \backslash \backslash bigr)\; \backslash cdot\; 3^$, admit more than one representation in this notation, as for example , that can be written as 0.1radix point
A decimal separator is a symbol used to separate the integer part from the fractional part of a number written in decimal form (e.g., "." in 12.45). Different countries officially designate different symbols for use as the separator. The choi ...

is not 1 are the ones remaining after the first step.
The second step removes numbers of the form 0.01xxxx...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 contai ...

s of $\backslash mathcal$. The same is true for the left limit points of $\backslash mathcal$, e.g. = 0.1222...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 contai ...

of $\backslash mathcal$ 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 $\backslash mathcal$ 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 $\backslash mathcal$ (but not dense in , 1 and makes up a countably infinite set. The numbers in $\backslash mathcal$ 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 $\backslash mathcal$ 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
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...

representation of a real number. In a formula,
:$f\; \backslash bigg(\; \backslash sum\_\; a\_k\; 3^\; \backslash bigg)\; =\; \backslash sum\_\; \backslash frac\; 2^$ where $\backslash forall\; k\backslash in\; \backslash N\; :\; a\_k\; \backslash in\; \backslash \; .$
For any number ''y'' in ,1 its binary representation can be translated into a ternary representation of a number ''x'' in $\backslash mathcal$ 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...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 num ...

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...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 ...

by Tarski), this provides another demonstration that $,\; \backslash mathcal,\; =\backslash mathfrak$.
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 conjectured that all algebraic irrational numbers are normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...

. 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. 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 tohomeomorphism
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 isomor ...

: $T\_L(\backslash mathcal)\backslash cong\; T\_R(\backslash mathcal)\backslash cong\; \backslash mathcal=T\_L(\backslash mathcal)\backslash cup\; T\_R(\backslash mathcal).$
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. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set $\backslash $ together with function composition 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.
Monoid ...

, the dyadic monoid
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 fractiona ...

.
The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, the Cantor set is a homogeneous space in the sense that for any two points $x$ and $y$ in the Cantor set $\backslash mathcal$, there exists a homeomorphism $h:\backslash mathcal\backslash to\; \backslash mathcal$ 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 $\backslash $. Then the map $h:\backslash ^\backslash N\backslash to\backslash ^\backslash N$ defined by $h\_n(u):=u\_n+x\_n+y\_n\; \backslash mod\; 2$ 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 $d\_f=\backslash ln(2)/\backslash ln(3)$ is thefractal 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 $x\_1,\; x\_2,\; \backslash ldots,\; x\_$ then the $d\_f$th moment is $x\_1^+x\_2^+\backslash cdots+x\_^=1$ since $x\_1=x\_2=\; \backslash cdots\; =x\_=1/3^n$.
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 anytopological 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 po ...

that is homeomorphic (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-clas ...

of a union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...

of open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

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 “size� ...

, the Heine–Borel theorem 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
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...

.
For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number with a ternary numeral of only 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
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 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 point, limit points of S, also known as the derived set ...

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 mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...

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
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 ...

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 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 ( , ), 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 m ...

, the Cantor set is an example of a Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in ...

.
As a topological space, the Cantor set is naturally homeomorphic to the product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...

of countably many copies of the space $\backslash $, 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 top ...

. 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 calle ...

s in two digits
:$2^\backslash mathbb\; =\; \backslash ,$
which can also be identified with the set of 2-adic integers. The basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...

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-s ...

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'' 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. 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 trans ...

.
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 set ...

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 $2^\backslash mathbb$: given two sequences $(x\_n),(y\_n)\backslash in\; 2^\backslash mathbb$, the distance between them is $d((x\_n),(y\_n))\; =\; 2^$, where $k$ is the smallest index such that $x\_k\; \backslash ne\; y\_k$; 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 "the ...

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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...

metric spaces, since any compact metric space is a continuous image of the Cantor set; however this construction is not unique and so the Cantor set is not universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
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** Universal TV, a ...

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 o ...

, where it is sometimes known as the ''representation theorem for compact metric spaces''.
For any integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...

''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 = ZMeasure and probability

The Cantor set can be seen as thecompact 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. 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 speakin ...

, 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
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 as ...

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 (or a set of first category) as a subset of ,1(although not as a subset of itself, since it is aBaire 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 e ...

). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set $\backslash mathbb\backslash cap;\; href="/html/ALL/l/,1.html"\; ;"title=",1">,1$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 $(1-f)^n\backslash to\; 0$ as $n\backslash to\backslash infty$ for any $f$ such that $0\backslash leq\; 1\; math>.\; On\; the\; other\; hand,\; "fat\; Cantor\; sets"\; of\; positive\; measure\; can\; be\; generated\; by\; removal\; of\; smaller\; fractions\; of\; the\; middle\; of\; the\; segment\; in\; each\; iteration.\; Thus,\; one\; can\; construct\; sets\; homeomorphic\; to\; the\; Cantor\; set\; that\; have\; positive\; Lebesgue\; measure\; while\; still\; being\; nowhere\; dense.\; If\; an\; interval\; of\; length$ r^n$($ r\backslash leq\; 1/3$)\; is\; removed\; from\; the\; middle\; of\; each\; segment\; at\; the\; \text{\'}\text{\'}n\text{\'}\text{\'}th\; iteration,\; then\; the\; total\; length\; removed\; is$ \backslash sum\_^\backslash infty\; 2^r^n=r/(1-2r)$,\; and\; the\; limiting\; set\; will\; have\; a;\; href="/html/ALL/l/Lebesgue\_measure.html"\; ;"title="Lebesgue\; measure">Lebesgue\; measure$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–Vol ...

, 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 :$\backslash frac\; =-\backslash frac\; c(x,t)\; +\; 2\backslash int\_x^\backslash infty\; (y-x)c(y,t)\; \backslash ,\; dy,$ and for the stochastic dyadic Cantor set :$=-xc(x,t)+(1+p)\backslash int\_x^\backslash infty\; c(y,t)\; \backslash ,\; dy,$ where $c(x,t)dx$ 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 $\backslash ln\; (1+p)/\backslash ln\; 2$. In the case of stochastic dyadic Cantor set the solution for $c(x,t)$ exhibitsdynamic scaling Dynamic scaling (sometimes known as Family-Vicsek scaling) is a litmus test that shows whether an evolving system exhibits self-similarity. In general a function is said to exhibit dynamic scaling if it satisfies:
:f(x,t)\sim t^\theta \varphi \left ...

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 ($\backslash int\; x^\; c(x,t)\; \backslash ,\; dx\; =\; \backslash text$) 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 of the Cantor set with itself, making it aMenger 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 $\backslash mathcal\; C$ as an example "of a perfect point-set, which is not everywhere-dense in any interval, however small." Cantor described $\backslash mathcal\; C$ in terms of ternary expansions, as "the set of all real numbers given by the formula: $z=c\_1/3\; +c\_2/3^2\; +\; \backslash cdots\; +\; c\_\backslash nu/3^\backslash nu\; +\backslash cdots$where the coefficients $c\_\backslash nu$ 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 $\backslash mathcal\; C$, 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 oftrigonometric series
In mathematics, a trigonometric series is a infinite series of the form
: \frac+\displaystyle\sum_^(A_ \cos + B_ \sin),
an infinite version of a trigonometric polynomial.
It is called the Fourier series of the integrable function f if the term ...

. The latter did much to set him on the course for developing an abstract, general theory of infinite sets.
See also

* 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–Vol ...

*Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. ...

*Cantor cube
In mathematics, a Cantor cube is a topological group of the form ''A'' for some index set ''A''. Its algebraic and topological structures are the group direct product and product topology over the cyclic group of order 2 (which is itself given th ...

*Antoine's necklace
In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeom ...

*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 Curv ...

*Knaster–Kuratowski fan
In topology, a branch of mathematics, the Knaster–Kuratowski fan (named after Polish mathematicians Bronisław Knaster and Kazimierz Kuratowski) is a specific connected topological space with the property that the removal of a single point ...

*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 illus ...

*Moser–de Bruijn sequence
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4, or equivalently the numbers whose binary representations are nonzero o ...

Notes

References

* * * * * . *External links

*Cantor Sets

an

Cantor Set and Function

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Cantor Set

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