Cantor Set: Difference between revisions
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==Cantor Ternary Set== | |||
A Cantor ternary set <math>C</math> of base-3 can be constructed through the infinite process of removing the middle one third of the open intervals from each closed interval composing the previous constructing sets sequentially. | |||
Specifically, starting from a closed interval <math>C_0 = [0,1]</math>, one can remove firstly the middle one third open interval, <math> (1/3,2/3)</math>, and get the remaining union of closed intervals <math> C_1 = C_0 \setminus (1/3,2/3)= [0,1/3] \cup [2/3,1] </math>. Then one can define <math>C2</math> with a similar manner: <math>C_2 = C_1 \setminus ((1/9,2/9)\cup(7/9,8/9)) = [0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1]</math>. | |||
Consecutively, each <math>C_n</math> is constructed by removing the middle one third of the closed intervals of <math>C_{n-1}</math>. The Cantor set <math>C</math> is then defined as follows.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §1.5 </ref><ref name="Craig">Craig, Katy. ''MATH 201A HW 5''. UC Santa Barbara, Fall 2020.</ref> | |||
:<math> C = \cup_{n=1}^{+\infty} C_n. </math> | |||
===Properties of Cantor Sets=== | |||
A Cantor set <math>C</math> constructed with the iterating process above has the following properties.<ref name="Folland">Gerald B. Folland, ''Real Analysis: Modern Techniques and Their Applications, second edition'', §1.5 </ref> | |||
* <math>C</math> is compact, nowhere dense, and totally disconnected. Moreover, <math>C</math> has no isolated points. | |||
* Denote <math>\lambda</math> as the Lebesgue measure and <math>\mathcal{B}_{\mathbb{R}}</math> as the Borel set defined on <math>\mathbb{R}</math>. Then <math>C</math> is measurable, and <math>\lambda(C) = 0</math>. | |||
==Cantor Function== | |||
The Cantor set can be used to define [[Cantor function]]. | |||
==References== | |||
Revision as of 23:12, 5 December 2020
Cantor Ternary Set
A Cantor ternary set of base-3 can be constructed through the infinite process of removing the middle one third of the open intervals from each closed interval composing the previous constructing sets sequentially. Specifically, starting from a closed interval , one can remove firstly the middle one third open interval, , and get the remaining union of closed intervals . Then one can define with a similar manner: . Consecutively, each is constructed by removing the middle one third of the closed intervals of . The Cantor set is then defined as follows.[1][2]
![{\displaystyle C_{0}=[0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74582af582e1f60ff548ee41c59f5d38f7023032)
![{\displaystyle C_{1}=C_{0}\setminus (1/3,2/3)=[0,1/3]\cup [2/3,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87823b13ad376ae01b5ba1e907fc8c8c921088ab)
![{\displaystyle C_{2}=C_{1}\setminus ((1/9,2/9)\cup (7/9,8/9))=[0,1/9]\cup [2/9,1/3]\cup [2/3,7/9]\cup [8/9,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b33dffe9df476a93f738bb965f37df5917fe46b9)
Properties of Cantor Sets
A Cantor set constructed with the iterating process above has the following properties.[1]
- is compact, nowhere dense, and totally disconnected. Moreover, has no isolated points.
- Denote as the Lebesgue measure and as the Borel set defined on . Then is measurable, and .
Cantor Function
The Cantor set can be used to define Cantor function.