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Math Help - Cantor set questions

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    Cantor set questions

    Defining a "Cantor space" as a non-empty, complete, totally disconnected perfect set C\subseteq[0,1] with d(x,y)=|x-y| in R (if I'm right this would only be THE Cantor set (i.e. infintiely removing all mittle thirds fron [0,1]) up to homomorphism, either way we don't "officially" know this yet) prove or disprove:

    (i) finite unions of Cantor sets are also Cantor sets.
    (ii)countable unions of Cantor sets are also Cantor sets.
    (iii)the intersection of two Cantor sets is either empty or also a Cantor set.
    (iv)every Cantor set is a Lebesque null set.
    (v)the complement of a Cantor set is never a Lebesque null set.
    (vi) every infinite, closed, totally disconnected set A\subseteq[0,1] is uncountable.
    (vii) every Cantor set is uncountable.
    (viii)every non-empty, complete, perfect set A\subseteq[0,1] is uncountable.

    I'm pretty sure (i) is true and (ii) is false (due to the fact that Q is countable). If I'm right, (v) follows directly from (iv), which I'm pretty sure is true. I think there are simple counterexamples for (vi) and (viii) but I'm at a loss for ideas of what they would be, and (vii) would be trivial if I could use that statement up there in parenthesis.

    Please help!!!
    Last edited by realpart1/2; June 10th 2009 at 09:26 PM.
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