Cantor set

Not to be confused with Cantor space.

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith[1][2][3][4] and mentioned by German mathematician Georg Cantor in 1883.[5][6]

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 (,[5] Anmerkungen zu §10, /p. 590).

More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero dimensional.[7]

Expansion of a Cantor set. Each point in the set is represented here by a vertical line.
  1. ^ Smith, Henry J.S. (1874). "On the integration of discontinuous functions". Proceedings of the London Mathematical Society. First series. 6: 140–153.
  2. ^ The "Cantor set" was also discovered by Paul du Bois-Reymond (1831–1889). See du Bois-Reymond, Paul (1880), "Der Beweis des Fundamentalsatzes der Integralrechnung", Mathematische Annalen (in German), 16, footnote on p. 128. The "Cantor set" was also discovered in 1881 by Vito Volterra (1860–1940). See: Volterra, Vito (1881), "Alcune osservazioni sulle funzioni punteggiate discontinue" [Some observations on point-wise discontinuous function], Giornale di Matematiche (in Italian), 19: 76–86.
  3. ^ Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser Verlag. pp. 162–165. ISBN 9783034850513.
  4. ^ Stewart, Ian (26 June 1997). Does God Play Dice?: The New Mathematics of Chaos. Penguin. ISBN 0140256024.
  5. ^ a b Cantor, Georg (1883). "Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets), Part 5]. Mathematische Annalen (in German). 21: 545–591. doi:10.1007/bf01446819. S2CID 121930608. Archived from the original on 2015-09-24. Retrieved 2011-01-10.
  6. ^ Peitgen, H.-O.; Jürgens, H.; Saupe, D. (2004). Chaos and Fractals: New Frontiers of Science (2nd ed.). N.Y., N.Y.: Springer Verlag. p. 65. ISBN 978-1-4684-9396-2.
  7. ^ Kechris, Alexander S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics. Vol. 156. Springer New York, NY. pp. 31, 35. doi:10.1007/978-1-4612-4190-4. ISBN 978-0-387-94374-9.