Cantor Set: Difference between revisions
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==Cantor Ternary Set== | ==Cantor Ternary Set== | ||
A Cantor ternary set <math>C</math> of base-3 can be constructed from <math>[0,1]</math> through the iterative process of removing the open middle third from each closed interval. Specifically, starting from a closed interval <math>C_0 = [0,1]</math>, one can first remove the open middle third, <math> (1/3,2/3)</math>, to 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>. Next, one repeat the process of removing open middle thirds from each closed interval, ie <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>. | A Cantor ternary set <math>C</math> of base-3 can be constructed from <math>[0,1]</math> through the iterative process of removing the open middle third from each closed interval. Specifically, starting from a closed interval <math>C_0 = [0,1]</math>, one can first remove the open middle third, <math> (1/3,2/3)</math>, to 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>. Next, one repeat the process of removing open middle thirds from each closed interval, ie <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>. | ||
Each <math>C_n</math> is then constructed iteratively by removing the middle one third | Each <math>C_n</math> is then constructed iteratively by removing the middle one third from each closed intervals of <math>C_{n-1}</math>. The Cantor set <math>C</math> is then obtained when one repeats the process infinitely many times, or equivalently:<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 = \ | <math> C = \bigcap_{n=1}^{+\infty} C_n. </math> | ||
===Properties of Cantor Sets=== | ===Properties of Cantor Sets=== | ||
A Cantor set <math>C</math> | A Cantor set <math>C</math> 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. | * <math>C</math> is closed, compact, nowhere dense, and totally disconnected. Moreover, <math>C</math> has no isolated points. | ||
* | * <math>C</math> is Lebesgue measurable, with Lebesgue measure <math>\lambda(C) = 0</math>. | ||
* Cantor set is in bijection with <math>[0,1]</math>, giving us a counterexample of a noncountable set having zero measure. | * Cantor set is in bijection with <math>[0,1]</math>, giving us a counterexample of a noncountable set having zero measure. | ||
Latest revision as of 03:15, 18 December 2020
Cantor Ternary Set
A Cantor ternary set of base-3 can be constructed from through the iterative process of removing the open middle third from each closed interval. Specifically, starting from a closed interval , one can first remove the open middle third, , to get the remaining union of closed intervals . Next, one repeat the process of removing open middle thirds from each closed interval, ie . Each is then constructed iteratively by removing the middle one third from each closed intervals of . The Cantor set is then obtained when one repeats the process infinitely many times, or equivalently:[1][2]
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\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 has the following properties.[1]
- is closed, compact, nowhere dense, and totally disconnected. Moreover, has no isolated points.
- is Lebesgue measurable, with Lebesgue measure .
- Cantor set is in bijection with , giving us a counterexample of a noncountable set having zero measure.
Cantor Function
The Cantor set can be used to define Cantor Function, an increasing function which is continuous but has zero derivative almost everywhere.