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 October 27th, 2013, 12:42 PM #1 Newbie   Joined: Feb 2012 Posts: 5 Thanks: 0 Cantor Set Equivalence of Definitions Hey guys, I have an assignment for my coursework and I don't even know where to start. "Examine at least two different definitions of the middle-thirds Cantor set from the literature, and prove that they define the same set." If anybody who's familiar with this topic, could you direct me to some relevant literature, give some tips, or even bounce ideas, I'd be very grateful. Please, spare the responses of type "just google it" Thanks
 October 27th, 2013, 03:40 PM #2 Member   Joined: May 2013 Posts: 31 Thanks: 3 Re: Cantor Set Equivalence of Definitions Do you guess that the question mean Cantor set defined by the following? [...] the Cantor ternary set is built by removing the middle thirds of a line segment is an example of a more general idea that of a perfect set that is nowhere dense." http://en.wikipedia.org/wiki/Cantor_set I ask because is possible construct Cantor-like set removing open middle intervals of other lengths.
 October 28th, 2013, 01:05 AM #3 Newbie   Joined: Feb 2012 Posts: 5 Thanks: 0 Re: Cantor Set Equivalence of Definitions Yes question is specifically about cantor set obtained by removing middle lengths. (but I guess can be extended to general case of removing any length) This is my idea so far: Use definition of cantor set as a limit of intersections (something of this sort http://www.proofwiki.org/wiki/Definitio ... ersections ) and another definition as a ternary representation. I think I could manage proving their equivalence starting from ternary and showing that it is the same as limit of intersection, but a good proof requires showing the equivalence both ways. Any ideas on how to show equivalence starting with limits of intersections and arriving at ternary representation? Any input would be appreciated Also please ask if anything needs clarification.
 October 28th, 2013, 02:01 PM #4 Member   Joined: May 2013 Posts: 31 Thanks: 3 Re: Cantor Set Equivalence of Definitions Then, you can do the following: Given $0\: <\: \beta \:< \:1$, remove from $[0,\: 1]$ an open interval $\left(1/2 - \frac{\beta}{4},\: 1/2 + \frac{\beta}{4}\right)$ and denote by $E_{1}$ the union of the two remaining closed intervals. Next remove the open middle intervals of length $\frac{\beta}{2^3}$ of the two intervals consisting $E_{1}$ and denote by $E_{2}$ the union of four remaining closed intervals. Repeat the process with each of four intervals, removing the open middle intervals of length $\frac{\beta}{2^5}$ . Continuing the process, we obtain the sequence ${E_{n}}$ of sets, where $E_{n}$ is the union of$2^n$ closed intervals, and we put $A= \underset{n}{\cap} E_{n}$. This set A, is the Cantor-like set.
October 28th, 2013, 02:02 PM   #5
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Re: Cantor Set Equivalence of Definitions

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 Originally Posted by Prokhartchin Then, you can do the following: Given $0\: <\: \beta \:< \:1$, remove from $[0,\: 1]$ an open interval $\left(\frac{1}{2} - \frac{\beta}{4},\: \frac{1}{2} + \frac{\beta}{4}\right)$ and denote by $E_{1}$ the union of the two remaining closed intervals. Next remove the open middle intervals of length $\frac{\beta}{2^3}$ of the two intervals consisting $E_{1}$ and denote by $E_{2}$ the union of four remaining closed intervals. Repeat the process with each of four intervals, removing the open middle intervals of length $\frac{\beta}{2^5}$ . Continuing the process, we obtain the sequence ${E_{n}}$ of sets, where $E_{n}$ is the union of$2^n$ closed intervals, and we put $A= \underset{n}{\cap} E_{n}$. This set A, is the Cantor-like set.

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