# 康托尔集

## 康托尔集的构造

${\displaystyle C_{0}:=[0,1]}$

${\displaystyle C_{n}:={\frac {C_{n-1}}{3}}\cup \left({\frac {2}{3}}+{\frac {C_{n-1}}{3}}\right)={\frac {1}{3}}\left(C_{n-1}\cup (2+C_{n-1})\right)}$, 对于${\displaystyle n\geq 1}$

${\displaystyle {\mathcal {C}}:=}$ ${\displaystyle \lim _{n\to \infty }C_{n}}$ ${\displaystyle =\bigcap _{n=0}^{\infty }C_{n}=\bigcap _{n=m}^{\infty }C_{n}}$, 对于 ${\displaystyle m\geq 0}$.

## 註釋

1. ^ Georg Cantor (1883) "Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets)]，Mathematische Annalen, vol. 21, pages 545–591.
2. ^ H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science 2nd ed. (N.Y., N.Y.: Springer Verlag, 2004), page 65.
3. ^ Henry J.S. Smith (1875) “On the integration of discontinuous functions.” Proceedings of the London Mathematical Society, Series 1, vol. 6, pages 140–153.
4. ^ “康托尔集”还由Paul du Bois-Reymond发现（1831–1889）。参见：Paul du Bois-Reymond (1880) “Der Beweis des Fundamentalsatzes der Integralrechnung，” Mathematische Annalen, vol. 16, pages 115–128的第128页的脚注。“康托尔集”还由Vito Volterra在1881年发现（1860–1940）。参见：Vito Volterra (1881) “Alcune osservazioni sulle funzioni punteggiate discontinue” [Some observations on point-wise discontinuous functions]，Giornale di Matematiche, vol. 19, pages 76–86.
5. ^ José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Basel, Switzerland: Birkhäuser Verlag, 1999), pages 162–165.
6. ^ Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos
7. ^ Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006.
8. ^ Mohsen Soltanifar, A Different Description of A Family of Middle-a Cantor Sets, American Journal of Undergraduate Research, Vol 5, No 2, pp 9–12, 2006.