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Thread: Both open and closed set proof

  1. #1
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    Both open and closed set proof

    Prove that the only sets that are both open and closed are \mathbb{R} and the empty set \phi.

    How do you go about proving that every other set is either open, closed, or neither? There are infinitely many other sets.
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  2. #2
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    What do you know about connected sets? The space \mathbb{R} is connected.
    If F\subseteq \mathbb{R} , is non-trivial, and is both open and closed then F\cup F^c=\mathbb{R}.
    Is that a contradiction? WHY?
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  3. #3
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    If F is open then F^c is closed. So if a set is both open and closed, what does that make the complement of that set? Neither open nor closed?
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  4. #4
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    No. These are known as clopen sets (both open and closed).
    So if F is clopen then F^c is clopen.
    There is a well known result about connected sets and clopen sets.
    Otherwise, I have not seen this topic discussed.
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  5. #5
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    So if I can prove that the empty set, \phi, is both open and closed (or clopen) then I can say that the set of all reals, \mathbb{R}, is clopen because \mathbb{R} is the complement to \phi?
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  6. #6
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    It is completely obvious that those two sets are both open and closed.
    They are the only two sets in \mathbb{R} that are.
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