Editing Cantor set (section)

{{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.]]