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Question about Cantor Set

  1. Jul 2, 2008 #1
    Let C be the Cantor set
    Let A be the set which is the union of those end points of each interval in each step of the cantor set construction

    It seems to be true that A is countable and C is uncountable. Moreover, A is a proper subset of C. But I cannot imagin what kind of the points in C - A should be, for if p is not an end point of some interval, p seems to be an interior point of some interval but contor set contains no segment. Is there any way to understand these points besides using an ternary expansion to prove that C is uncountable and hence ponits like this simply exist?

    Any help would be appreciated
  2. jcsd
  3. Jul 2, 2008 #2

    matt grime

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    C is the *limit* of that construction. There is nothing odd about an uncountable set being a limit of a sequence of countable ones like that. The real numbers would be such an example: take the set D_n to be the numbers with n digits after the decimal point. The 'limit' of these sets are the real numbers.

    The things in C-A are the limit points of sequences in A.
  4. Jul 2, 2008 #3
    ah, I understand it now. Thanks. sometimes it is difficult to have an explicit view of the existence of limit points
  5. Jul 4, 2008 #4


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    Alternatively, the points of the Cantor set are numbers 0 <= x <=1 which can be written in base 3 with no 1's. eg x=0.20022002...
    The endpoints are such numbers which eventually become repeating 2s or repeating 0s. eg, x = 0.2022222...
    You can approximate any number as closely as you like by ones with a terminating base-3 expansion.
  6. Jul 9, 2008 #5
    I saw you solved already your doubts.
    I remember having the same doubt when I saw the Cantor dust for the first time.
    It took quite lot of time to understand it by myself!
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