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{{Short description|Set of points on a line segment}}
{{Distinguish|Cantor space}}
In [[mathematics]], the '''Cantor set''' is a [[set (mathematics)|set]] of points lying on a single [[line segment]] that has a number of unintuitive properties.
Through consideration of this set, Cantor and others helped lay the foundations of modern [[point-set topology]].
More generally, in topology, a [[Cantor space]] is a topological space [[Homeomorphism|homeomorphic]] to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally [[Homeomorphism|homeomorphic]] to the countable product <math>{\underline 2}^{\N}</math> of the [[discrete two-point space]] <math>\underline 2 </math>. By a theorem of [[L. E. J. Brouwer]], this is equivalent to being perfect, nonempty, compact, [[Metrizable space|metrizable]] and zero-dimensional.<ref name=":0">{{Cite book |last=Kechris |first=Alexander S. |url=https://link.springer.com/book/10.1007/978-1-4612-4190-4 |title=Classical Descriptive Set Theory |series=Graduate Texts in Mathematics |publisher=Springer New York, NY |year=1995 |volume=156 |isbn=978-0-387-94374-9 |pages=31, 35 |language=en |doi=10.1007/978-1-4612-4190-4}}</ref>
[[File:Cantor Zoom.gif|center|thumb|600px|Zoom in Cantor set. Each point in the set is represented here by a vertical line.]]▼
[[File:Cantor Set Expansion.gif|center|thumb|600px|[[File:Cantor set binary tree.svg|400px|class=skin-invert]]
▲
==Construction and formula of the ternary set==
The Cantor ternary set <math>\mathcal{C}</math> is created by iteratively deleting the [[open interval|''open'']] middle third from a set of line segments. One starts by deleting the open middle third <math display="inline">\left(\frac{1}{3}, \frac{2}{3}\right)</math> from the [[interval (mathematics)|interval]] <math>\textstyle\left[0, 1\right]</math>, leaving two line segments: <math display="inline">\left[0, \frac{1}{3}\right]\cup\left[\frac{2}{3}, 1\right]</math>. Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: <math display="inline">\left[0, \frac{1}{9}\right]\cup\left[\frac{2}{9}, \frac{1}{3}\right]\cup\left[\frac{2}{3}, \frac{7}{9}\right]\cup\left[\frac{8}{9}, 1\right]</math>.
The Cantor ternary set contains all points in the interval <math>[0,1]</math> that are not deleted at any step in this [[ad infinitum|infinite process]]. The same
: <math>C_0 := [0,1]</math>
and
: <math>C_n := \frac{C_{n-1}} 3 \cup \left(\frac 2 {3} +\frac{C_{n-1}} 3 \right) = \frac13 \bigl(C_{n-1} \cup \left(2 + C_{n-1} \right)\bigr)
for <math>n \ge 1
: <math>
The first six steps of this process are illustrated below.
[[Image:Cantor set in seven iterations.svg|729px|class=skin-invert|
Cantor ternary set, in seven iterations]] Using the idea of self-similar transformations, <math>T_L(x)=x/3,</math>
: <math>
where every middle third is removed as the open interval <math display="inline">\left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}}\right)
: <math>
where the middle third <math display="inline">\left(\frac{3k+1}{3^n},\frac{3k+2}{3^n}\right) </math> of the foregoing closed interval <math display="inline">\left[\frac{k+0}{3^{n-1}},\frac{k+1}{3^{n-1}}\right] = \left[\frac{3k+0}{3^n},\frac{3k+3}{3^n}\right]
This process of removing middle thirds is a simple example of a [[finite subdivision rule]]. The complement of the Cantor ternary set is an example of a [[fractal string]].
[[File:Cantor set binary tree.svg|400px]]
In arithmetical terms, the Cantor set consists of all [[real
=== Mandelbrot's construction by "curdling" ===
In ''[[The Fractal Geometry of Nature]]'', mathematician [[Benoit Mandelbrot]] provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of <math>\mathcal{C}</math>. His narrative begins with imagining a bar, perhaps of lightweight metal, in which the bar's matter "curdles" by iteratively shifting towards its extremities. As the bar's segments become smaller, they become thin, dense slugs that eventually grow too small and faint to see.<blockquote>CURDLING: The construction of the Cantor bar results from the process I call curdling. It begins with a round bar. It is best to think of it as having a very low density. Then matter "curdles" out of this bar's middle third into the end thirds, so that the positions of the latter remain unchanged. Next matter curdles out of the middle third of each end third into its end thirds, and so on ad infinitum until one is left with an infinitely large number of infinitely thin slugs of infinitely high density. These slugs are spaced along the line in the very specific fashion induced by the generating process. In this illustration, curdling (which eventually requires hammering!) stops when both the printer's press and our eye cease to follow; the last line is indistinguishable from the last but one: each of its ultimate parts is seen as a gray slug rather than two parallel black slugs.<ref name=":3" /></blockquote>
== Composition ==
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:<math>\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.</math>
So that the proportion left is <math>1
This calculation suggests that the Cantor set cannot contain any
It may appear that ''only'' the endpoints of the construction segments are left, but that is not the case either. The number {{sfrac|1|4}}, for example, has the unique ternary form 0.020202... = {{overline|0.|02}}. It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of {{sfrac|1
| last1 = Belcastro | first1 = Sarah-Marie
| last2 = Green | first2 = Michael
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:<math> \left\{x \in [0,1] \mid \exists i \in \N_0: x \, 3^i \in \Z \right\} \qquad \Bigl(\subset \N_0 \, 3^{-\N_0} \Bigr) </math>
which is a [[countably infinite]] set.
As to [[cardinality]], [[almost all]] elements of the Cantor set are not endpoints of intervals, nor [[rational number|rational]] points like {{sfrac|1
== 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 set|uncountable]]. To see this, we show that there is a [[function (mathematics)|function]] ''f'' from the Cantor set <math>\mathcal{C}</math> to the closed interval <math>[0, 1]</math> that is [[Surjective function|surjective]] (i.e. ''f'' maps from <math>\mathcal{C}</math> onto <math>[0, 1]</math>) so that the
To construct this function, consider the points in the <math>[0,
When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...<sub>3</sub> where xxxxx...<sub>3</sub> is strictly between 00000...<sub>3</sub> and 22222...<sub>3</sub>. So the numbers remaining after the first step consist of
* Numbers of the form 0.0xxxxx...<sub>3</sub> (including 0.022222...<sub>3</sub> = 1/3)
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It is worth emphasizing that numbers like 1, {{sfrac|1|3}} = 0.1<sub>3</sub> and {{sfrac|7|9}} = 0.21<sub>3</sub> are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...<sub>3</sub> = {{overline|0.|2}}<sub>3</sub>, {{sfrac|1|3}} = 0.0222...<sub>3</sub> = {{overline|0.0|2}}<sub>3</sub> and {{sfrac|7|9}} = 0.20222...<sub>3</sub> = {{overline|0.20|2}}<sub>3</sub>.
All the latter numbers are
This set of endpoints is [[
The function from <math>\mathcal{C}</math> to <math>[0, 1]</math> is defined by taking the ternary numerals that do consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a [[Binary numeral system#Representing real numbers|binary]] representation of a real number. In a formula,
:<math>f \bigg( \sum_{k\in \N} a_k 3^{-k} \bigg) = \sum_{k\in \N} \frac{a_k}{2} 2^{-k}</math> where <math>\forall k\in \N : a_k \in \{0,2\} .</math>
For any number ''y'' in <math>[0, 1]</math>, its binary representation can be translated into a ternary representation of a number ''x'' in <math>\mathcal{C}</math> by replacing all the 1s by 2s. With this, ''f''(''x'') = ''y'' so that ''y'' is in the [[Range of a function|range]] of ''f''. For instance if ''y'' = {{sfrac|3|5}} = 0.100110011001...<sub>2</sub> = {{overline|0.|1001}}, we write ''x'' = {{overline|0.|2002}} = 0.200220022002...<sub>3</sub> = {{sfrac|7|10}}. Consequently, ''f'' is surjective. However, ''f'' is ''not'' [[injective function|injective]] — the values for which ''f''(''x'') coincides are those at opposing ends of one of the ''middle thirds'' removed. For instance, take
:{{sfrac|1|3}} = {{overline|0.0|2}}<sub>3</sub> (which is a right limit point of <math>\mathcal{C}</math> and a left limit point of the middle third [{{sfrac|1|3}}, {{sfrac|2|3}}])
:{{sfrac|2|3}} = {{overline|0.2|0}}<sub>3</sub> (which is a left limit point of <math>\mathcal{C}</math> and a right limit point of the middle third [{{sfrac|1|3}}, {{sfrac|2|3}}])
so
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& {}^1\!\!/\!_2
\end{array}</math>
Thus there are as many points in the Cantor set as there are in the interval <math>[0,
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
It has been
=== Self-similarity ===
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Repeated [[iterated function|iteration]] of <math>T_L</math> and <math>T_R</math> 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 <math>\{T_L, T_R\}</math> together with [[function composition]] forms a [[monoid]], the [[dyadic monoid]].
The [[automorphism]]s 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 <math>x</math> and <math>y</math> in the Cantor set <math>\mathcal{C}</math>, there exists a homeomorphism <math>h:\mathcal{C}\to \mathcal{C}</math> with <math>h(x)=y</math>. An explicit construction of <math>h</math> can be described more easily if we see the Cantor set [[
=== 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]] 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 (set theory)|complement]] of a [[union (set theory)|union]] of [[open set]]s, it itself is a [[closed set|closed]] subset of the reals, and therefore a [[complete
For any point in the Cantor set and any arbitrarily small [[neighborhood (mathematics)|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]]. 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 set|nowhere dense]] in the interval.
Every point of the Cantor set is also an accumulation point of the
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]]. Consequently, the Cantor set is [[totally disconnected]]. As a compact totally disconnected [[Hausdorff space]], the Cantor set is an example of a [[Stone space]].
As a
:<math>2^\mathbb{N} = \{(x_n) \mid x_n \in \{0,1\} \text{ for } n \in \mathbb{N}\},</math>
which can also be identified with the set of [[p-adic
From the above characterization, the Cantor set is [[Homeomorphism|homeomorphic]] to the [[p-adic
The Cantor set is a subset of the reals, which are a [[metric space]] with respect to the [[absolute difference|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|''p''-adic metric]] on <math>2^\mathbb{N}</math>: given two sequences <math>(x_n),(y_n)\in 2^\mathbb{N}</math>, the distance between them is <math>d((x_n),(y_n)) = 2^{-k}</math>, where <math>k</math> is the smallest index such that <math>x_k \ne y_k</math>; 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 [[topological space|topology]] on the Cantor set.
We have seen above that the Cantor set is a totally disconnected [[perfect set|perfect]] compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is [[Homeomorphism|homeomorphic]] to the Cantor set. See [[Cantor space]] for more on spaces [[Homeomorphism|homeomorphic]] to the Cantor set.
The Cantor set is sometimes regarded as "universal" in the [[
For any [[integer]] ''q'' ≥ 2, the topology on the [[group (mathematics)|group]] G = '''Z'''<sub>''q''</sub><sup>ω</sup> (the countable direct sum) is discrete.<!-- I don't know how the Z_q^w should be LaTeX-ified. --> Although the [[Pontrjagin dual]] Γ is also '''Z'''<sub>''q''</sub><sup>ω</sup>, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is [[Homeomorphism|homeomorphic]] to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case ''q'' = 2. (See Rudin 1962 p 40.)
===Measure and probability===
The Cantor set can be seen as the [[compact group]] 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]], 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.<ref>{{cite web | url=http://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | title=Theorem 36: the Cantor set is an uncountable set with zero measure | first=Laura | last=Irvine | website=Theorem of the week | access-date=2012-09-27 | archive-url=https://web.archive.org/web/20160315212203/https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/ | archive-date=2016-03-15 | url-status=dead }}</ref> In contrast, the set has a [[Hausdorff measure]] of <math>1</math> in its dimension of <math>\log_3(2)</math>.<ref>
{{cite book |last=Falconer |first=K. J. |date=July 24, 1986 |title=The Geometry of Fractal Sets |url=http://mate.dm.uba.ar/~umolter/materias/referencias/1.pdf |pages=14–15 |publisher=Cambridge University Press |isbn=9780521337052}}
</ref>
===Cantor numbers===
If we define a Cantor number as a member of the Cantor set, then<ref>{{cite book | title=Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise | first=Manfred | last=Schroeder | publisher=Dover | date=1991 | pages=164–165 | isbn=0486472043}}</ref>
# Every real number in <math>[0, 2]</math> is the sum of two Cantor numbers.
# Between any two Cantor numbers there is a number that is not a Cantor number.
=== Descriptive set theory ===
The Cantor set is a [[
== Variants ==
[[File:cantor_set_radial.svg|thumb|Radial plot of the first ten steps<ref>{{cite web | url=http://gist.github.com/curran/74cb4d255acf072633a2df0d9b9be7c3 | title=Radial Cantor Set }}</ref>]]
===Smith–Volterra–Cantor set===
{{main|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 {{sfrac|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 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 <math>(1-f)^n\to 0</math> as <math>n\to\infty</math> for any
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 [[Homeomorphism|homeomorphic]] to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length <math>r^n</math> (<math>r\leq 1/3</math>) is removed from the middle of each segment at the ''n''th iteration, then the total length removed is <math display="inline">\sum_{n=1}^\infty 2^{n-1}r^n=r/(1-2r)</math>, and the limiting set will have a [[Lebesgue measure]] of <math>\lambda=(1-3r)/(1-2r)</math>. Thus, in a sense, the middle-thirds Cantor set is a limiting case with <math>r=1/3</math>. If <math>0<r<1/3</math>, then the remainder will have positive measure with <math>0<\lambda<1</math>. The case <math>r=1/4</math> is known as the [[Smith–Volterra–Cantor set]], which has a Lebesgue measure of <math>1/2</math>.
=== Cantor dust ===
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==Historical remarks==
[[File:Cantor dust in two dimensions iteration 2.svg|thumb|an image of the 2nd iteration of Cantor dust in two dimensions]][[File:Cantor dust in two dimensions iteration 4.svg|alt=an image of the 4th iteration of Cantor dust in two dimensions|thumb|an image of the 4th iteration of Cantor dust in two dimensions]]
[[File:Cantor-like Column Capital Ile de Philae Description d'Egypte 1809.jpg|thumb|Column capital with pattern evocative of the Cantor set, but expressed in binary rather than ternary. Engraving of Île de Philae from ''Description d'Égypte'' by Jean-Baptiste Prosper Jollois and Édouard Devilliers, Imprimerie Impériale, Paris, 1809-1828]]▼
Cantor introduced what we call today the Cantor ternary set <math>\mathcal C</math> as an example "of a [[Perfect set|perfect point-set]], which is not everywhere-dense in any interval, however small."<ref name=":1">{{Cite web |last=Cantor |first=Georg |date=2021 |title="Foundations of a general theory of sets: A mathematical-philosophical investigation into the theory of the infinite", English translation by James R Meyer |url=https://www.jamesrmeyer.com/infinite/cantor-grundlagen.html#Fn_22_a |access-date=2022-05-16 |website=www.jamesrmeyer.com |at=Footnote 22 in Section 10}}</ref><ref name=":2">{{Cite journal |last=Fleron |first=Julian F. |date=1994 |title=A Note on the History of the Cantor Set and Cantor Function |url=https://www.jstor.org/stable/2690689 |journal=Mathematics Magazine |volume=67 |issue=2 |pages=136–140 |doi=10.2307/2690689 |jstor=2690689 |issn=0025-570X}}</ref> Cantor described <math>\mathcal C</math> in terms of ternary expansions, as "the set of all real numbers given by the formula: <math>z=c_1/3 +c_2/3^2 + \cdots + c_\nu/3^\nu +\cdots </math>where the coefficients <math>c_\nu</math> arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements."<ref name=":1" />
A topological space <math>P</math> is perfect if all its points are limit points or, equivalently, if it coincides with its [[Derived set (mathematics)|derived set]] <math>P'</math>. Subsets of the real line, like <math>\mathcal C</math>, can be seen as topological spaces under the induced subspace topology.<ref name=":0" />
[[Benoit Mandelbrot]] wrote much on Cantor dusts and their relation to [[Fractals in nature|natural fractals]] and [[statistical physics]].<ref name=":3">{{Cite book |last=Mandelbrot |first=Benoit B. |title=The fractal geometry of nature |date=1983 |isbn=0-7167-1186-9 |edition=Updated and augmented |___location=New York |oclc=36720923}}</ref> He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. In [[The Fractal Geometry of Nature]], he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the [[Koch snowflake|Koch]] and [[Peano curve]]s," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming <math>\mathcal C</math> to be interesting in science."<ref name=":3" />
▲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 [[Fourier series|trigonometric series]] might fail to converge. The discovery did much to set him on the course for developing an [[axiomatic set theory|abstract, general theory of infinite sets]].
== See also ==
[[File:Cantor dust in two dimensions iteration 6.svg|alt=an image of the 6th iteration of Cantor dust in two dimensions|thumb|an image of the 6th iteration of Cantor dust in two dimensions]]
*[[Classification of discontinuities#
*[[Smith–Volterra–Cantor set]]
*[[Cantor function]]
*[[Cantor cube]]
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*[[Knaster–Kuratowski fan]]
*[[List of fractals by Hausdorff dimension]]
▲*[[Moser–de Bruijn sequence]][[File:Cantor-like Column Capital Ile de Philae Description d'Egypte 1809.jpg|thumb|Column capital with pattern evocative of the Cantor set, but expressed in binary rather than ternary. Engraving of Île de Philae from ''Description d'Égypte'' by Jean-Baptiste Prosper Jollois and Édouard Devilliers, Imprimerie Impériale, Paris, 1809-1828]]
==Notes==
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