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Cantor set

  1. Aug 19, 2012 #1

    bedi

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    Why there is no interior points in a Cantor set? Please explain me in detail.
     
    bedi, Aug 19, 2012
  2. jcsd
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  3. Aug 19, 2012 #2

    micromass

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    What did you try already?
     
    micromass, Aug 19, 2012
  4. Aug 19, 2012 #3

    bedi

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    I've been watching a lecture and didn't understand why there wouldn't be any point whose neighbourhood is completely surrounded by the cantor set. Oh I think I start to grasp it now, since every interval's "middle" is removed and that process goes on forever, every point's neighbourhood becomes somewhat "incomplete". Am I wrong?
     
    bedi, Aug 19, 2012
  5. Aug 19, 2012 #4

    lavinia

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    What is the measure of the Cantor set?
     
    lavinia, Aug 19, 2012
  6. Aug 19, 2012 #5

    bedi

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    I don't know that measure thing yet...
     
    bedi, Aug 19, 2012
  7. Aug 19, 2012 #6

    lavinia

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    OK. Tr describing the Canor set in the trinary system.
     
    lavinia, Aug 19, 2012
  8. Aug 19, 2012 #7

    Bacle2

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    Notice that like Lavinia said,the terms in the (Standard) Cantor set C have no 1's in

    their base-3 expansion. Now try to show,given x in C --so that there are no

    1's in the decimal expansion of x -- that, no matter how close you go about x in

    (x-e,x +e ) , you will hit a number y in (x-e,x+e) ,whose decimal expansion _does_

    have a 1 in it . Hint: you can cut-off the decimal expansion of x at any point,

    as far back as you want.
     
    Bacle2, Aug 19, 2012
  9. Aug 20, 2012 #8

    bedi

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    Thank to all of you. But 1/3 is included in the cantor set and in trinary it's 0.1 isn't it? I see there is no finite trinary number in the set but why is 0.1 included?

    edit: Now I think I'm okay. In the trinary number system, 1/3 can be written not only as 0.1 but also as 0.222222...
     
    Last edited: Aug 20, 2012
    bedi, Aug 20, 2012
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