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Math Help - prove that D is dense

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    prove that D is dense

    Prove that a subset D of a metric space M is dense in M iff D \bigcap U nonempty for every nonempty open set U \subset M

    so given  D \bigcap U nonempty we need to show that cl(D)=M, how does this work?
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    Re: prove that D is dense

    Quote Originally Posted by wopashui View Post
    Prove that a subset D of a metric space M is dense in M iff D \bigcap U nonempty for every nonempty open set U \subset M

    so given  D \bigcap U nonempty we need to show that cl(D)=M, how does this work?
    Here's the basic duality that needs to be noticed D\cap U=\varnothing implies D\subseteq (M-U) and M-U is closed and not the full space (since U\ne\varnothing). Get it?
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    Re: prove that D is dense

    Quote Originally Posted by wopashui View Post
    Prove that a subset D of a metric space M is dense in M iff D \bigcap U nonempty for every nonempty open set U \subset M so given  D \bigcap U nonempty we need to show that cl(D)=M, how does this work?
    Here is a slightly different discussion of this.
    The statement that x is a contact point of D means that x\in D or x is a limit point of D [i.e. x\in cl(D)].

    So you are asked to show that each point of M is a contact point of D.

    Suppose that t\in M but t\notin cl(D). Then there is a ball \mathcal{B}(t;\delta) that contains no point of D.
    WHY? And why is that a contradiction?
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