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Math Help - Open/closed sets of Real Numbers

  1. #1
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    Open/closed sets of Real Numbers

    For a set contained in the real numbers R, If R - S is open, then S is closed.

    I have no idea where to start on this!
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  2. #2
    Member Sylvia104's Avatar
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    Re: Open/Closed sets of real numbers

    Let x be a real number. If x\notin S, use the fact that \mathbb R\setminus S is open to show that x is not a limit point of S. It will then follow that if x is a limit point of S then x\in S, proving that S contains all its limit points and so is closed.
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  3. #3
    Member ModusPonens's Avatar
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    Re: Open/closed sets of Real Numbers

    Another way is to note that R-(R-S)=S, that is S is the complement of an open set. It helps to think better, sometimes, to write X-A as A^c. You will see the DeMorgan laws quicker as well as other stuff.
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