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Math Help - The Cantor Set

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    The Cantor Set

    how would one show that the interior of the cantor set is the empty set?
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    Quote Originally Posted by squarerootof2 View Post
    how would one show that the interior of the cantor set is the empty set?
    In my view the fastest way to prove that is to show that each point of the Cantor Set is a boundary point.
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    Quote Originally Posted by squarerootof2 View Post
    how would one show that the interior of the cantor set is the empty set?
    You can't because the cantor set is nonempty compact set of measure zero.

    I will not show this but... let E be the cantor set then

    x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x
    has a ternary expansion whose digits
    b_k \ne 1 \mbox{ }\forall \mbox{ }k
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    Quote Originally Posted by TheEmptySet View Post
    You can't because the cantor set is nonempty compact set of measure zero.

    I will not show this but... let E be the cantor set then

    x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x
    has a ternary expansion whose digits
    b_k \ne 1 \mbox{ }\forall \mbox{ }k
    Sorry my mistake

    I misread the interior

    Sorry.
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    Quote Originally Posted by squarerootof2 View Post
    how would one show that the interior of the cantor set is the empty set?
    If the Cantor set had a non-empty interior, then it would contain an open interval and hence would have positive measure.
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