康托尔集

在数学中，康托尔集，由德国数学家格奥尔格·康托尔在1883年引入[1][2]（但由亨利·约翰·斯蒂芬·史密斯在1875年发现[3][4][5][6]），是位于一条线段上的一些点的集合，具有许多显著和深刻的性质。通过考虑这个集合，康托尔和其他数学家奠定了现代点集拓扑学的基础。虽然康托尔自己用一种一般、抽象的方法定义了这个集合，但是最常见的构造是康托尔三分点集，由去掉一条线段的中间三分之一得出。康托尔自己只附带介绍了三分点集的构造，作为一个更加一般的想法——一个无处稠密的完备集的例子。

カントール集合のような模様がついた柱頭。Jollois, Jean-Baptiste Prosper; Devilliers, Edouard, , Paris: Imprimerie Imperiale, 1809-1828 请检查|date=中的日期值 (帮助) よりフィラエ島の彫刻

康托尔集的构造

康托尔集是由不断去掉线段的中间三分之一而得出。首先从区间 $[0,1]$ 中去掉中间的三分之一（ ${\frac {1}{3}},{\frac {2}{3}}$ ），留下两条线段： $[0,{\frac {1}{3}}]\cup [{\frac {2}{3}},1]$ 。然后，把这两条线段的中间三分之一都去掉，留下四条线段： $[0,{\frac {1}{9}}]\cup [{\frac {2}{9}},{\frac {1}{3}}]\cup [{\frac {2}{3}},{\frac {7}{9}}]\cup [{\frac {8}{9}},1]$ 。把这个过程一直进行下去，其中第 $n$ 个集合为：

{\frac {C_{n-1}}{3}}\cup \left({\frac {2}{3}}+{\frac {C_{n-1}}{3}}\right).

康托尔集就是由所有过程中没有被去掉的区间 $[0,1]$ 中的点组成。

下面的图显示了这个过程的最初六个步骤。

有些学术论文详细描述了康托尔集的明确公式。[7][8]

参见

康托尔函数
康托尔立方体
谢尔宾斯基地毯
科赫雪花
门格海绵
分形列表

註釋

Georg Cantor (1883) "Über unendliche, lineare Punktmannigfaltigkeiten V" [On infinite, linear point-manifolds (sets)]，Mathematische Annalen, vol. 21, pages 545–591.
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.
Henry J.S. Smith (1875) “On the integration of discontinuous functions.” Proceedings of the London Mathematical Society, Series 1, vol. 6, pages 140–153.
“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.
José Ferreirós, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (Basel, Switzerland: Birkhäuser Verlag, 1999), pages 162–165.
Ian Stewart, Does God Play Dice?: The New Mathematics of Chaos
Mohsen Soltanifar, On A sequence of cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9, 2006.
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.

參考文獻

Steen, Lynn Arthur; Seebach, J. Arthur Jr., Dover reprint of 1978, Berlin, New York: Springer-Verlag, 1995 [1978], ISBN 978-0-486-68735-3, MR 507446 (See example 29).
Gary L. Wise and Eric B. Hall, Counterexamples in Probability and Real Analysis. Oxford University Press, New York 1993. ISBN 0-19-507068-2. (See chapter 1).
cut-the-knot上的康托尔集
cut-the-knot上的康托尔集与函数

外部連結

Cantor Sets and Cantor Set and Function at cut-the-knot

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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.