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Encyclopedia > Cantor set

The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. This article is in need of attention from an expert on the subject. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

The Cantor set is defined by repeatedly removing the middle thirds of line segments. One starts by removing the middle third from the unit interval [0, 1], leaving [0, 1/3] ∪ [2/3, 1]. Next, the "middle third" of each of the remaining intervals is removed. This process is continued ad infinitum. The Cantor set consists of all points in the interval [0, 1] that are not removed at any step in this infinite process. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Ad infinitum is a Latin phrase meaning to infinity. ...

The first six steps of this process are illustrated below. Image File history File links Cantor_set_in_seven_iterations. ...

## What's in the Cantor set? GA_googleFillSlot("encyclopedia_square");

Since the Cantor set is defined as the set of points not excluded, the proportion of the unit interval remaining can be found by total length removed. This total is the geometric series In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... $sum_{n=0}^infty frac{2^n}{3^{n+1}} = frac{1}{3} + frac{2}{9} + frac{4}{27} + frac{8}{81} + cdots = frac{1}{3}left(frac{1}{1-frac{2}{3}}right) = 1.$

So that the proportion left is 1 – 1 = 0. Alternatively, it can be observed that each step leaves 2/3 of the length in the previous stage, so that the amount remaining is 2/3 × 2/3 × 2/3 × ..., an infinite product with a limit of 0.

From the calculation, it may seem surprising that there would 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 (1/3, 2/3) from the original interval [0, 1] leaves behind the points 1/3 and 2/3. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...

### The Cantor set contains no measureable intervals

The nth iteration of the algorithm that produces the Cantor set yields 2n intervals, each of them of length 3n. Since the actual Cantor set is the infinite intersection of these iterations, it follows that the intervals shrink to zero size as n approaches infinity. It is a trivial property of the real numbers that an interval of any length is uncountable by Cantor's diagonalization argument (referenced in the next section). One of the most fascinating properties of the Cantor set is that it is uncountable.

## Properties

### The Cantor set is uncountable

It can be shown that there are as many points left behind in this process as there were that were removed. To see this, we show that there is a function f from the Cantor set C to the closed interval [0,1] that is surjective (i.e. f maps from C onto [0,1]) so that the cardinality of C is no less than that of [0,1]. Since C is a subset of [0,1], its cardinality is also no greater, so the two cardinalities must in fact be equal. A surjective function. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...

To construct this function, consider the points in the [0, 1] interval in terms of base 3 (or ternary) notation. In this notation, 1/3 can be written as 0.13 and 2/3 can be written as 0.23, so the middle third (to be removed) 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 consists of Ternary or trinary is the base-3 numeral system. ...

• Numbers of the form 0.0xxxxx...
• 1/3 = 0.13 = 0.022222...3 (This alternative "recurring" representation of a number with a terminating numeral occurs in any positional system.)
• 2/3 = 0.122222...3 = 0.23
• Numbers of the form 0.2xxxxx...3

All of which can be stated as those numbers with a ternary numeral 0.0xxxxx...3 or 0.2xxxxx...3 A numeral is a symbol or group of symbols that represents a number. ...

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 whose first two digits are not 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it will not to be excluded at any step, it must have a numeral consisting entirely of 0's and 2's. It is worth emphasising that numbers like 1, 1/3 = 0.13 and 7/9 = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0's and 2's: 1 = 0.2222...3, 1/3 = 0.022222...3 and 7/9 = 0.2022222...3. So while a number in C may have either a terminating or a recurring ternary numeral, only one of its numerals consists entirely of 0's and 2's.

The function from C to [0,1] is defined by taking the numeral that does consist entirely of 0's and 2's, and replacing all the 2's by 1's. In a formula, $f left ( sum_{k=1}^infty a_k 3^{-k} right ) = sum_{k=1}^infty (a_k/2) 2^{-k}$

For any number y in [0,1], its binary representation can be translated into a ternary representation of a number x in C by replacing all the 1's by 2's. With this, f(x) = y so that y is in the range of f. For instance if y=3/5=0.100110011001...2, we write x = 0.200220022002...3 = 7/10. Consequently f is surjective; however, f is not injective — interestingly enough, the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, 7/9 = 0.2022222...3 and 8/9 = 0.2200000...3 so f(7/9) = 0.101111...2 = 0.113 = f(8/9). In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

In other words, the "endpoints" of the Cantor set are all numbers with ternary representation consisting of only 0's and 2's. Since there is a clear bijection between the ternary numbers consisting of only the digits 0 and 2 and the binary numbers consisting of the digits 0 and 1, it follows that the number of endpoints in the Cantor set is equal to the number of binary strings. The number of binary strings is uncountable by Cantor's diagonal argument, thus the Cantor set contains an uncountable number of points, though it contains no interval. This is the "paradox" of the Cantor set, that it contains as many points as the interval from which it is taken, yet it itself contains no interval. Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ... In mathematics the term countable set is used to describe the size of a set, e. ...

### The Cantor set is a fractal

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 1/3 and translated. Its Hausdorff dimension is equal to ln(2)/ln(3). It can be formed by intersecting a Sierpinski carpet with any of its lines of reflectional symmetry (such as reading the center scanline). The boundary of the Mandelbrot set is a famous example of a fractal. ... A self-similar object is exactly or approximately similar to a part of itself. ... In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, âˆž]) associated to any metric space . ... The Sierpinski carpet is a plane fractal first described by WacÅ‚aw SierpiÅ„ski. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

### Topological and analytical properties

As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since the Cantor set is the complement 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 bounded, the Heine-Borel theorem says that it must be compact. In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Ã‰mile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

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 0's and 2's, as well as numbers whose ternary numerals contain 1's. Hence, every point in the Cantor set is an accumulation point, but none is an interior point. A closed set in which every point is an accumulation point is also called a perfect set in topology, while a closed subset of the interval with no interior points is nowhere dense in the interval. In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ... In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ... In mathematics a derived set is a construction in point-set topology that consists of taking the set of limit points of a given subset S of a topological space X. The derived set of S is usually denoted by S′. A subset S of a topological space X is... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ...

For two points in the Cantor set, there will be some ternary digit where they differ — one d 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. Consequently the Cantor set is totally disconnected. As a compact totally disconnected Hausdorff space, the Cantor set is an example of a Stone space. 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 natural topology induced from that of X called the subspace topology (or the relative topology). ... In topology, a clopen set (or closed-open set) in a topological space is a set which is both open and closed. ... In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Boolean spaces, i. ...

As a topological space, the Cantor set is homeomorphic to the product of countably many copies of the space {0, 1}, where each copy carries the discrete topology, as can easily be shown using the ternary expansion used to prove its uncountability. The basis for the open sets of the product topology are the cylinder sets; the homeomorphism maps these to the subspace topology that the Cantor set inherits from the natural topology on the real number line. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics the term countable set is used to describe the size of a set, e. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

The Cantor set is a homogeneous space in the sense that for any two points x and y in the Cantor set C, there exists a homeomorphism f : CC with f(x) = y. In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

The Cantor set can be characterized by these properties: every nonempty totally-disconnected perfect compact metric space is homeomorphic to the Cantor set. See Cantor space for more on spaces homeomorphic to the Cantor set. In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ...

The Cantor set is "universal in the category of compact metric spaces". This means that any compact metric space is a continuous image of the Cantor set. This fact has important applications in functional analysis. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

Every point of the Cantor set is a cluster point of the Cantor set. Every point of the Cantor set is also a cluster point of the complement of the Cantor set. In mathematics, informally speaking, a limit point (or cluster point) of a set S in a topological space X is a point x in X that can be approximated by points of S other than x as well as one pleases. ...

## Variants of the Cantor set

See main article 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. The resulting sets are all homeomorphic to the Cantor set and also have Lebesgue measure 0. In the case where the middle 8/10 of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0's and 9's. In mathematics, the Smith-Volterra-Cantor set is a set of points on the real line satisfying the following interesting combination of properties: SVC is nowhere dense (in particular it contains no intervals), SVC has positive measure. ...

By removing progressively smaller percentages of the remaining pieces in every step, one can also construct sets homeomorphic to the Cantor set that have positive Lebesgue measure, while still being nowhere dense. See Smith-Volterra-Cantor set for an example. In topology, a subset A of a topological space X is called nowhere dense if the interior of the closure of A is empty. ... In mathematics, the Smith-Volterra-Cantor set is a set of points on the real line satisfying the following interesting combination of properties: SVC is nowhere dense (in particular it contains no intervals), SVC has positive measure. ...

## Historical remarks

This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge. The discovery did much to set him on the course for developing an abstract, general theory of infinite sets. Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...

## Sonification of the Cantor set

Interpreting the Cantor set as a time series, it can be sonified. This is a test of how well one can perceive temporal self-similar structures by listening. Here is an example of the 11th iteration, with a minimal trigger signal time difference of 0.042 ms: . Sonification is the use of non-speech audio to convey information or perceptualize data. ...

In mathematics, the Cantor function is a function c : [0,1] → [0,1] defined as follows: Express x in base 3. ... In mathematics, the term Cantor space is sometimes used to denote the topological abstraction of the classical Cantor set: A topological space is a Cantor space if it is homeomorphic to the Cantor set. ... The first four iterations of the Koch snowflake. ... Menger sponge, created by using IFS. Iterated function systems or IFS, are a kind of fractal that was conceived in its present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. ... Cantor dust, named after the mathematician Georg Cantor, is the two-dimensional version of the Cantor set. ... In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. ...

## Historical references

• Georg Cantor, On the Power of Perfect Sets of Points (De la puissance des ensembles parfait de points), Acta Mathematica 2 (1884) English translation reprinted in Classics on Fractals, ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7

## Modern references

• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition). (See example 29).

Counterexamples in Topology (1970) is a mathematics book by topologists Lynn A. Steen and J. Arthur Seebach, Jr. ... Results from FactBites:

 Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints (509 words) He distinguished between countable and uncountable sets, and was able to prove that the set of all rational numbers Q is countable, while the set off all real numbers R is uncountable, and therefore, though both were infinite, R was strictly larger. The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set. Cantor came came to the conclusion that the Absolute was beyond man's reach, and identified this concept with God.
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