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Math Help - Non-empty, compact, disconnected and limit points

  1. #1
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    Non-empty, compact, disconnected and limit points

    I am at the moment trying to get through some basic set theory and I'm getting very stuck with the proofs. This question is from a textbook I am studying from and as it is a prove question there is no solution in the back
    Any help with explanations would be very useful

    Let Non-empty, compact, disconnected and limit points-mathforum.jpg where E0 = [0, 1] and Ek is constructed by removing the
    second, fourth and sixth sevenths of each component Ek-1. Each set that is removed is open.

    I need to show that F is non-empy, compact, totally disconnected and that 0 is a limit point of F
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  2. #2
    Super Member girdav's Avatar
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    Re: Non-empty, compact, disconnected and limit points

    Show that each E_k is compact, which will give you the first two points. For totally disconnectedness, show that E doesn't contain any interval. Show that for each k there is x_k such that |x_k|\leq 2^{-1}\cdot 7^{-k}.
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