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Math Help - metric space

  1. #1
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    metric space

    let (X,d) be a metric space and A\neq \phi a subset of X then  x \in \overline{A}  iff
     d(x,A)=0
    Last edited by flower3; May 24th 2009 at 01:42 AM.
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  2. #2
    Super Member Gamma's Avatar
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    Quote Originally Posted by flower3 View Post
    let (X,d) be a metric space and A\neq \phi a subset of X then  x \in \overline{A}  iff
     d(x,A)=0

    (\Rightarrow )
    Suppose x \in \overline{A} but d(x,A)=\delta >0. Then consider B(x,\frac{\delta}{2})this is an open neighborhood containing x, but does not intersect A, thus x \not \in \overline{A}, contradiction. d(x,A)=0 as desired.

    (\Leftarrow) Let d(x,A)=0. Suppose x \not \in \overline{A} then there exists an open set U containing x, but does not intersect A. By definition of open set, this means there is a basis element (a ball) containing x and contained in U. That is x \in B(x,\delta) \subset U \subset A^c. But this means d(x,A)>\delta >0 contradiction. thus x \in \overline{A} as desired.

    QED
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