Editing Cantor set

{{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. It was discovered in 1874 by [[Henry John Stephen Smith]]<ref>{{cite journal | first=Henry J.S. | last=Smith | date=1874 | title=On the integration of discontinuous functions | journal=Proceedings of the London Mathematical Society | series=First series | volume=6 | pages=140–153| url=https://zenodo.org/record/1932560 }}</ref><ref>The "Cantor set" was also discovered by [[Paul du Bois-Reymond]] (1831–1889). See {{cite journal | at=footnote on p. 128 | first=Paul | last=du Bois-Reymond | date=1880 | url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002245256 | title=Der Beweis des Fundamentalsatzes der Integralrechnung | journal=Mathematische Annalen | volume=16 | language=de | mode=cs2}}.  The "Cantor set" was also discovered in 1881 by Vito Volterra (1860–1940).  See: {{cite journal | first=Vito | last=Volterra | date=1881 | title=Alcune osservazioni sulle funzioni punteggiate discontinue | trans-title=Some observations on point-wise discontinuous function | journal=Giornale di Matematiche | volume=19 | pages=76–86 | language=it | mode=cs2}}.</ref><ref>{{cite book | first=José | last=Ferreirós | title=Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics | url=https://archive.org/details/labyrinthofthoug0000ferr | url-access=registration | location=Basel, Switzerland | publisher=Birkhäuser Verlag | date=1999 | pages=[https://archive.org/details/labyrinthofthoug0000ferr/page/162 162]–165 | isbn=9783034850513 }}</ref><ref>{{cite book | first=Ian | last=Stewart | author-link=Ian Stewart (mathematician) | title=Does God Play Dice?: The New Mathematics of Chaos | date=26 June 1997 | publisher=Penguin | isbn=0140256024}}</ref> and mentioned by German mathematician [[Georg Cantor]] in 1883.<ref name=Cantor>{{cite journal | first=Georg | last=Cantor | date=1883 | url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002247461 | title=Über unendliche, lineare Punktmannigfaltigkeiten V | trans-title=On infinite, linear point-manifolds (sets), Part 5 | journal=Mathematische Annalen | volume=21 | pages=545–591 | language=de | doi=10.1007/bf01446819 | s2cid=121930608 | access-date=2011-01-10 | archive-url=https://web.archive.org/web/20150924114632/http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002247461 | archive-date=2015-09-24 | url-status=dead }}</ref><ref>{{cite book | first1=H.-O. | last1=Peitgen | first2=H. | last2=Jürgens | first3=D. | last3=Saupe | title=Chaos and Fractals: New Frontiers of Science | url=https://archive.org/details/chaosfractals00hein | url-access=limited | edition=2nd | location=N.Y., N.Y. | publisher=Springer Verlag | date=2004 | page=[https://archive.org/details/chaosfractals00hein/page/n79 65] | isbn=978-1-4684-9396-2}}</ref>

Through consideration of this set, Cantor and others helped lay the foundations of modern [[point-set topology]]. The most common construction is the '''Cantor ternary set''', built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a [[perfect set]] that is [[Nowhere dense set|nowhere dense]].<ref name=Cantor/>

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 Set Expansion.gif|center|thumb|600px|[[File:Cantor set binary tree.svg|400px|class=skin-invert]]
Expansion of a Cantor set. Each point in the set is represented here by a vertical line.]]

==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 construction can be described recursively by setting
: <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)</math>
for <math>n \ge 1</math>, so that
: <math>\mathcal{C} :=</math> [[Set-theoretic limit#Monotone sequences|<math>{\color{Blue}\lim_{n\to\infty}C_n}</math>]] <math>= \bigcap_{n=0}^\infty C_n = \bigcap_{n=m}^\infty C_n </math> &thinsp; for any &thinsp; <math>m \ge 0</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>T_R(x)=(2+x)/3</math> and <math>C_n =T_L(C_{n-1})\cup T_R(C_{n-1}),</math> the explicit closed formulas for the Cantor set are<ref>{{cite journal | first=Mohsen | last=Soltanifar | title=A Different Description of A Family of Middle-a Cantor Sets | journal=American Journal of Undergraduate Research | volume=5 | issue=2 | pages=9–12 | date=2006 | doi=10.33697/ajur.2006.014| doi-access=free }}</ref>
: <math>\mathcal{C}=[0,1] \,\setminus\, \bigcup_{n=0}^\infty \bigcup_{k=0}^{3^n-1} \left(\frac{3k+1}{3^{n+1}},\frac{3k+2}{3^{n+1}} \right)\!,</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> from the [[closed interval]] <math display="inline">\left[\frac{3k+0}{3^{n+1}},\frac{3k+3}{3^{n+1}}\right] = \left[\frac{k+0}{3^n},\frac{k+1}{3^n}\right]</math> surrounding it, or
: <math>\mathcal{C}=\bigcap_{n=1}^\infty \bigcup_{k=0}^{3^{n-1}-1} \left( \left[\frac{3k+0}{3^n},\frac{3k+1}{3^n}\right] \cup \left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right] \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]</math> is removed by intersecting with <math display="inline">\left[\frac{3k+0}{3^n},\frac{3k+1}{3^n}\right] \cup \left[\frac{3k+2}{3^n},\frac{3k+3}{3^n}\right]\!.</math>

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 number]]s of the [[unit interval]] <math>[0,1]</math> that do not require the digit 1 in order to be expressed as a [[Ternary numeral system|ternary]] (base 3) fraction. As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep [[binary tree]], where the path turns left or right at each level according to which side of a deleted segment the point lies on. Representing each left turn with 0 and each right turn with 2 yields the ternary fraction for a point. ''Requiring'' the digit 1 is critical: <math display="inline">\frac{1}{3}</math>, which is included in the Cantor set, can be written as <math display="inline">0.1</math>, but also as <math display="inline">0.0\bar{2}</math>, which contains no 1 digits and corresponds to an initial left turn followed by infinitely many right turns in the binary tree.

=== Mandelbrot's construction by "curdling" ===
In ''[[The Fractal Geometry of Nature]]'', 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 ==
Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., [[Lebesgue measure|measure]]) of the unit interval remaining can be found by total length removed.  This total is the [[geometric progression]]

:<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 - 1 = 0</math>.

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]]s (sets that do not include their endpoints). So removing the line segment <math display="inline">\left(\frac{1}{3}, \frac{2}{3}\right)</math> from the original interval <math>[0, 1]</math> leaves behind the points {{sfrac|1|3}} and {{sfrac|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 set|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 {{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|3}}.<ref name="College">{{citation
 | last1 = Belcastro | first1 = Sarah-Marie
 | last2 = Green | first2 = Michael
 | date = January 2001
 | doi = 10.2307/2687224
 | issue = 1
 | journal = The College Mathematics Journal
 | page = 55
 | title = The Cantor set contains <math>\tfrac{1}{4}</math>? Really?
 | volume = 32| jstor = 2687224
 }}</ref>
All endpoints of segments are ''terminating'' ternary fractions and are contained in the set
:<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|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 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 cardinality of <math>\mathcal{C}</math> is no less than that of <math>[0, 1]</math>.  Since <math>\mathcal{C}</math> is a [[subset]] of <math>[0, 1]</math>, its cardinality is also no greater, so the two cardinalities must in fact be equal, by the [[Cantor–Bernstein–Schröder theorem]].

To construct this function, consider the points in the <math>[0, 1]</math> interval in terms of base 3 (or [[ternary numeral system|ternary]]) notation. Recall that the proper ternary fractions, more precisely: the elements of <math>\bigl(\Z \setminus \{0\}\bigr) \cdot 3^{-\N_0}</math>, admit more than one representation in this notation, as for example {{sfrac|1|3}}, that can be written as 0.1<sub>3</sub> = {{overline|0.1|0}}<sub>3</sub>, but also as 0.0222...<sub>3</sub> = {{overline|0.0|2}}<sub>3</sub>, and {{sfrac|2|3}}, that can be written as 0.2<sub>3</sub> = {{overline|0.2|0}}<sub>3</sub> but also as 0.1222...<sub>3</sub> = {{overline|0.1|2}}<sub>3</sub>.<ref>This alternative recurring representation of a number with a terminating numeral occurs in any [[Numeral system#Positional systems in detail|positional system]] with [[Absolute value (algebra)#Types of absolute value|Archimedean absolute value]].</ref>
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)
* Numbers of the form 0.2xxxxx...<sub>3</sub> (including 0.222222...<sub>3</sub> = 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...<sub>3</sub> and 0.21xxxx...<sub>3</sub>, 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, {{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 "endpoints", and these examples are right [[limit point]]s of <math>\mathcal{C}</math>. The same is true for the left limit points of <math>\mathcal{C}</math>, e.g. {{sfrac|2|3}} = 0.1222...<sub>3</sub> = {{overline|0.1|2}}<sub>3</sub> = {{overline|0.2|0}}<sub>3</sub> and {{sfrac|8|9}} = 0.21222...<sub>3</sub> = {{overline|0.21|2}}<sub>3</sub> = {{overline|0.22|0}}<sub>3</sub>. All these endpoints are ''proper ternary'' fractions (elements of <math>\Z \cdot 3^{-\N_0}</math>) of the form {{sfrac|''p''|''q''}}, where denominator ''q'' is a [[power of 3]] when the fraction is in its [[Irreducible fraction|irreducible]] form.<ref name="College"/> 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]] of <math>\mathcal{C}</math> 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 <math>\mathcal{C}</math> if it again its ternary expansion contains no 1's and "ends" in infinitely many recurring 2s.

This set of endpoints is [[dense set|dense]] in <math>\mathcal{C}</math> (but not dense in <math>[0, 1]</math>) and makes up a [[countably infinite]] set. The numbers in <math>\mathcal{C}</math> 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 <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> &nbsp; where &nbsp; <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}}])&nbsp; and
:{{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
:<math>\begin{array}{lcl}
f\bigl({}^1\!\!/\!_3 \bigr) = f(0.0\overline{2}_3) = 0.0\overline{1}_2 = \!\! & \!\! 0.1_2 \!\! & \!\! = 0.1\overline{0}_2 = f(0.2\overline{0}_3) = f\bigl({}^2\!\!/\!_3 \bigr) . \\
& \parallel \\
& {}^1\!\!/\!_2
\end{array}</math>
Thus there are as many points in the Cantor set as there are in the interval <math>[0, 1]</math> (which has the [[Uncountable set|uncountable]] cardinality {{nowrap|<math>\mathfrak{c} = 2^{\aleph_0}</math>).}} 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 {{sfrac|1|4}}, which can be written as 0.020202...<sub>3</sub> = {{overline|0.|02}} in ternary notation.  In fact, given any <math>a\in[-1,1]</math>, there exist <math>x,y\in\mathcal{C}</math> such that <math>a = y-x</math>.  This was first demonstrated by [[Hugo Steinhaus|Steinhaus]] in 1917, who [[mathematical proof|proved]], via a geometric argument, the equivalent assertion that <math>\{(x,y)\in\mathbb{R}^2 \mid y=x+a\} \; \cap \; (\mathcal{C}\times\mathcal{C}) \neq\emptyset</math> for every <math>a\in[-1,1]</math>.<ref>{{Cite book|title=Real Analysis|url=https://archive.org/details/realanalysis00caro_315|url-access=limited|last=Carothers|first=N. L.|publisher=Cambridge University Press|year=2000|isbn=978-0-521-69624-1|location=Cambridge|pages=[https://archive.org/details/realanalysis00caro_315/page/n41 31]–32}}</ref> Since this construction provides an injection from <math>[-1,1]</math> to <math>\mathcal{C}\times\mathcal{C}</math>, we have <math>|\mathcal{C}\times\mathcal{C}|\geq|[-1,1]|=\mathfrak{c}</math> as an immediate [[corollary]].  Assuming that <math>|A\times A|=|A|</math> for any infinite set <math>A</math> (a statement shown to be equivalent to the [[axiom of choice]] [[Tarski's theorem about choice|by Tarski]]), this provides another demonstration that <math>|\mathcal{C}|=\mathfrak{c}</math>.

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]]s have the same property, but the Cantor set has the additional property of being [[Closed set|closed]], so it is not even [[Dense set|dense]] in any interval, unlike the irrational numbers which are dense in every interval.

It has been [[conjecture]]d that all [[algebraic number|algebraic]] irrational numbers are [[normal number|normal]].  Since members of the Cantor set are not normal in base 3, this would imply that all members of the Cantor set are either rational or [[transcendental number|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, <math>T_L(x)=x/3</math> and <math>T_R(x)=(2+x)/3</math>, which leave the Cantor set invariant up to [[homeomorphism]]: <math>T_L(\mathcal{C})\cong T_R(\mathcal{C})\cong \mathcal{C}=T_L(\mathcal{C})\cup T_R(\mathcal{C}).</math>

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|as a product space]] of  countably many copies of the discrete space <math>\{0,1\}</math>. Then the map <math>h:\{0,1\}^\N\to\{0,1\}^\N </math> defined by <math>h_n(u):=u_n+x_n+y_n \mod 2</math> is an [[involution (mathematics)|involutive]] homeomorphism exchanging  <math>x</math> and <math>y</math>.

=== 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 metric space]]. Since it is also [[totally bounded]], the [[Heine–Borel theorem]] says that it must be [[compact space|compact]].

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 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]].  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 topological space, the Cantor set is naturally [[Homeomorphism|homeomorphic]] to the [[product topology|product]] of countably many copies of the space <math>\{0, 1\}</math>, where each copy carries the [[discrete topology]]. This is the space of all [[sequence]]s in two digits 
:<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 integer|2-adic integers]]. The [[basis (topology)|basis]] for the open sets of the [[product topology]] are [[cylinder set]]s; the homeomorphism maps these to the [[subspace topology]] that the Cantor set inherits from the natural topology on the [[real line]]. This characterization of the [[Cantor space]] as a product of compact spaces gives a second proof that Cantor space is compact, via [[Tychonoff's theorem]].

From the above characterization, the Cantor set is [[Homeomorphism|homeomorphic]] to the [[p-adic integer|''p''-adic integers]], and, if one point is removed from it, to the [[p-adic number|''p''-adic numbers]].

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 [[category (mathematics)|category]] of [[compact space|compact]] metric spaces, since any compact metric space is a [[continuous function (topology)|continuous]] [[image (mathematics)|image]] of the Cantor set; however this construction is not unique and so the Cantor set is not [[universal property|universal]] in the precise [[category theory|categorical]] sense. The "universal" property has important applications in [[functional analysis]], where it is sometimes known as the ''representation theorem for compact metric spaces''.<ref>{{cite book | first=Stephen | last=Willard | title=General Topology | publisher=Addison-Wesley | date=1968 | asin=B0000EG7Q0}}</ref>

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 [[meagre set]] (or a set of first category) as a subset of <math>[0, 1]</math> (although not as a subset of itself, since it is a [[Baire space]]).  The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide.  Like the set <math>\mathbb{Q}\cap[0,1]</math>, the Cantor set <math>\mathcal{C}</math> is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of <math>[0, 1]</math>.  However, unlike <math>\mathbb{Q}\cap[0,1]</math>, which is countable and has a "small" cardinality, <math>\aleph_0</math>, the cardinality of <math>\mathcal{C}</math> is the same as that of <math>[0, 1]</math>, the continuum <math>\mathfrak{c}</math>, and is "large" in the sense of cardinality.  In fact, it is also possible to construct a subset of <math>[0, 1]</math> that is meagre but of positive measure and a subset that is non-meagre but of measure zero:<ref>{{Cite book|title=Counterexamples in analysis|last=Gelbaum, Bernard R.|date=1964|publisher=Holden-Day|others=Olmsted, John M. H. (John Meigs Hubbell), 1911-1997|isbn=0486428753|location=San Francisco|oclc=527671}}</ref>  By taking the countable union of "fat" Cantor sets <math>\mathcal{C}^{(n)}</math> of measure <math>\lambda = (n-1)/n</math> (see Smith–Volterra–Cantor set below for the construction), we obtain a set <math display="inline">\mathcal{A} := \bigcup_{n=1}^{\infty}\mathcal{C}^{(n)}</math>which has a positive measure (equal to 1) but is meagre in [0,1], since each <math>\mathcal{C}^{(n)}</math> is nowhere dense.  Then consider the set <math display="inline">\mathcal{A}^{\mathrm{c}} = [0,1] \setminus\bigcup_{n=1}^\infty \mathcal{C}^{(n)}</math>.  Since <math>\mathcal{A}\cup\mathcal{A}^{\mathrm{c}} = [0,1]</math>, <math>\mathcal{A}^{\mathrm{c}}</math> cannot be meagre, but since <math>\mu(\mathcal{A})=1</math>, <math>\mathcal{A}^{\mathrm{c}}</math> must have measure zero.

== 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 <math>f</math> such that <math>0<f\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 [[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 ===
<!-- This is linked to by the redirect "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 a [[Cantor space]]. Like the Cantor set, Cantor dust has [[Measure zero|zero measure]].<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=46|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA46}}</ref>
[[File:Cantorcubes.gif|thumb|right|250px|[[Cantor cube]]s recursion progression towards Cantor dust]]
{|
|[[Image:Cantor dust.svg|thumb|'''Cantor dust''' (2D)]]
|[[Image:Cantors cube.jpg|thumb|'''Cantor dust''' (3D)]]
|}

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.<ref>{{cite book|author=Helmberg, Gilbert|title=Getting Acquainted With Fractals|publisher=Walter de Gruyter|year=2007|isbn=978-3-11-019092-2|page=48|url=https://books.google.com/books?id=PbrlYO83Oq8C&pg=PA48}}</ref> One 3D analogue of this is the [[Menger sponge]].

==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]]
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" />

Cantor was led to the study of derived sets by his results on uniqueness of [[Fourier series|trigonometric series]].<ref name=":2" /> The latter did much to set him on the course for developing an [[axiomatic set theory|abstract, general theory of infinite sets]].

[[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. |url=https://www.worldcat.org/oclc/36720923 |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" />

== 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#Examples|The indicator function of the Cantor set]]
*[[Smith–Volterra–Cantor set]]
*[[Cantor function]]
*[[Cantor cube]]
*[[Antoine's necklace]]
*[[Koch snowflake]]
*[[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==
{{Reflist|30em}}

==References==
{{refbegin}}
* {{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach Jr. | title=Counterexamples in Topology | orig-year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 | at=Example 29| title-link=Counterexamples in Topology }}
* {{cite book | first1=Gary L. | last1=Wise | first2=Eric B. | last2=Hall | title=Counterexamples in Probability and Real Analysis | url=https://archive.org/details/counterexamplesi0000wise | url-access=registration | publisher=[[Oxford University Press]] | location=New York | date=1993 | isbn=0-19-507068-2 | at=Chapter 1}}
* {{cite book | author-link=K. J. Falconer | first=K. J. | last=Falconer | title=Geometry of Fractal Sets | url=https://archive.org/details/geometryoffracta00falc | url-access=registration | publisher=[[Cambridge University Press]] | date=24 July 1986 | series=Cambridge Tracts in Mathematics | issue=85 | isbn=0521337054}}
* {{cite book | first=Pertti | last=Mattila  | title=Geometry of Sets and Measures in Euclidean Space: Fractals and rectifiability | publisher=Cambridge University Press | date=25 February 1999 | series=Cambridge studies in advanced mathematics | issue=44 | isbn=0521655951}}
* {{cite book | first=Pertti | last=Mattila | title=Fourier Analysis and Hausdorff Dimension | publisher=Cambridge University Press | date=2015 | series=Cambridge studies in advanced mathematics | issue=150 | isbn=9781316227619}}.
* {{cite book | first=A. | last=Zygmund | author-link=A. Zygmund | title=Trigonometric Series, Vols. I and II |title-link = Trigonometric Series | publisher=Cambridge University Press | date=1958}}
{{refend}}

==External links==
* {{springer|title=Cantor set|id=p/c020250}}
* [http://www.cut-the-knot.org/do_you_know/Cantor2.shtml Cantor Sets] and [http://www.cut-the-knot.org/do_you_know/cantor.shtml Cantor Set and Function] at [[cut-the-knot]]
* [https://platonicrealms.com/encyclopedia/Cantorn-set Cantor Set] at Platonic Realms

{{Fractals}}
{{Authority control}}

{{DEFAULTSORT:Cantor Set}}
[[Category:Measure theory]]
[[Category:Topological spaces]]
[[Category:Sets of real numbers]]
[[Category:Georg Cantor]]
[[Category:L-systems]]