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Math Help - if A is an open subset of R, then A is dense if and only if R - A is nowhere dense

  1. #1
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    if A is an open subset of R, then A is dense if and only if R - A is nowhere dense

    i am trying to prove the => direction first.

    assuming that A is dense, i know that  \forall (a, b) \subseteq \mathbb{R}, \exists x \in A \ s.t. \ x \in (a, b)

    i also know that A is open so  x \in A \Rightarrow \exists B_{r}(x) \subseteq A

    the trouble i am having is that i am not sure if B_{r}(x) \subseteq (a, b) because if that was true then it follows that  \mathbb{R} - A is nowhere dense.

    could someone help me continue the proof? thanks.
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    Re: if A is an open subset of R, then A is dense if and only if R - A is nowhere dens

    Since (a,b) is open, A\cap (a,b) is open and you can choose r such that B_r(x)\subset A\cap (a,b).
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    Re: if A is an open subset of R, then A is dense if and only if R - A is nowhere dens

    Quote Originally Posted by oblixps View Post
    i am trying to prove the => direction first.
    assuming that A is dense, i know that  \forall (a, b) \subseteq \mathbb{R}, \exists x \in A \ s.t. \ x \in (a, b)
    i also know that A is open so  x \in A \Rightarrow \exists B_{r}(x) \subseteq A
    the trouble i am having is that i am not sure if B_{r}(x) \subseteq (a, b) because if that was true then it follows that  \mathbb{R} - A is nowhere dense.
    A\cap (a,b) is an open set. Is it not?
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