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Math Help - Prove that A and B are separated

  1. #1
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    Prove that A and B are separated

    Fix
    p inX, r > 0, define A to be the set of all q inX for which

    d
    (p, q) < r, define B similarly, with > in place of <. Prove that A and

    B are separated.

    Wont the closure of A be { q in X | d(p,q) <=r}
    and closure of B be {q in X | d(p,q) >=r }
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  2. #2
    Senior Member roninpro's Avatar
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    Wont the closure of A be { q in X | d(p,q) <=r}
    and closure of B be {q in X | d(p,q) >=r }
    Not necessarily. If d is the discrete metric, and r=1, the statement fails.

    You might be able to show that the closure is contained in there, however.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by poorna View Post
    Fix
    p inX, r > 0, define A to be the set of all q inX for which

    d
    (p, q) < r, define B similarly, with > in place of <. Prove that A and

    B are separated.

    Wont the closure of A be { q in X | d(p,q) <=r}
    and closure of B be {q in X | d(p,q) >=r }
    Note that in a metric space \left(\mathcal{M},d\right) if E\subseteq\mathcal{M} then \text{cl}_\mathcal{M}\text{ }E=\left\{x\in\mathcal{M}:d(x,E)=0\right\}...so what?

    Also, I'm sure you've proven that if A,B are disjoint open subspaces of a metric space they're separated, right? So, note that \varphi:\mathcal{M}\to\mathbb{R}:x\mapsto d(x,p) is continuous and you're to sets are \varphi^{-1}((-\infty,r)),\varphi^{-1}((r,\infty)) so that they are disjoint open.
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