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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 $\displaystyle \mathbb{R}$ and the empty set $\displaystyle \phi$.

    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 $\displaystyle \mathbb{R}$ is connected.
    If $\displaystyle F\subseteq \mathbb{R} $, is non-trivial, and is both open and closed then $\displaystyle F\cup F^c=\mathbb{R}$.
    Is that a contradiction? WHY?
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  3. #3
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    If $\displaystyle F$ is open then $\displaystyle F^c$ 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 $\displaystyle F$ is clopen then $\displaystyle F^c$ 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, $\displaystyle \phi$, is both open and closed (or clopen) then I can say that the set of all reals, $\displaystyle \mathbb{R}$, is clopen because $\displaystyle \mathbb{R}$ is the complement to $\displaystyle \phi$?
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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 $\displaystyle \mathbb{R}$ that are.
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