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Let $C_0$ be the segment $[0,1]$ $C_2$ will be $[0,1]$, with the middle third, an open set removed, so $[0,1/3]\cup[2/3,1]$

First, if we removed closed sets would the cantor set, the limit of what remains from this process, be the same?

I was able to show that the limit of the sequence of partial sums of the pieces converges to $1$. And this happens if the removed bits are open or closed, it doesn't matter. Don't know if that helps or not.

Second, I've been trying for two hours and I still can't find a sensible way to write $C_n$, in some compact notation.

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    Not the same, but all the endpoints removed are rationals of the shape $k/3^n$ (however, not all such rationals are endpoints). Thus in your "remove closed intervals" version, we are removing only countably many more points than in the Cantor set construction. Since the Cantor set has the cardinality of the continuum, this means that in your version, "almost all" of the original Cantor set remains.2011-07-18
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    I want to edit the title - "acceding" is a typo, right? Is it supposed to be "ascending"? But then again, the question doesn't seem to be about any kind of order on the Cantor set "endpoints". Oh, and I see $C_0$ and $C_2$, but what happened to $C_1$?2011-07-19
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    "Acceding" was edited to "accending"? The world is a strange place sometimes.2011-07-19
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    OK, I got impatient and edited the title, and while I was there I did away with the descriptive-set-theory tag. OP can take care of $C_1$, or not.2011-07-19
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    @Gerry: Why did you remove the [descriptive-set-theory] tag?2011-07-19
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    @Asaf, descriptive-set-theory is a technical term - it doesn't just mean, "questions about describing sets." Having said that, I may have been too hasty. http://en.wikipedia.org/wiki/Descriptive_set_theory says "Descriptive set theory begins with the study of Polish spaces and their Borel sets," and gives the Cantor set as an example of a Polish space. If someone who, unlike me, actually knows something about descriptive set theory wants to put the tag back, I'll get out of the way.2011-07-20
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    @Gerry: Seeing how I just finished taking a course in descriptive set theory, I figured I had the basic judgment skills whether this tag relates to my answer or not :-) I will put it back now, if you don't mind.2011-07-20

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