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Math Help - [SOLVED] Analysis problem

  1. #1
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    [SOLVED] Analysis problem

    a point metric space X.show {a}^c is open.
    Last edited by oaza; December 11th 2008 at 09:40 PM.
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  2. #2
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    Quote Originally Posted by oaza View Post
    Can someone help me with that proof.
    Let x be any point in a metric space X.Prove that {x}^c(complement of x) is open.
    I have the same problem to solve and also I do not know how to approach that.
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  3. #3
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    Quote Originally Posted by oaza View Post
    Can someone help me with that proof.
    Quote Originally Posted by oaza View Post
    Let x be any point in a metric space X.Prove that {x}^c(complement of x) is open.
    Come on you two guys, this is such a fundamental property of metrics.
    The distance between two points in a metric space is positive.
    y \in \left\{ x \right\}^c \Rightarrow \quad d(x,y) > 0
    \left\{ x \right\}^c = \bigcup\limits_{y \in \left\{ x \right\}^c } {B\left( {y;d(x,y)} \right)}
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  4. #4
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    That is really it?!!
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  5. #5
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    Quote Originally Posted by Plato View Post
    Come on you two guys, this is such a fundamental property of metrics.
    The distance between two points in a metric space is positive.
    y \in \left\{ x \right\}^c \Rightarrow \quad d(x,y) > 0
    \left\{ x \right\}^c = \bigcup\limits_{y \in \left\{ x \right\}^c } {B\left( {y;d(x,y)} \right)}
    I always think proofs are supposed to be so long...lol is it really that simple?
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  6. #6
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    Quote Originally Posted by megamet2000 View Post
    is it really that simple?
    That shows that the complement is the union of open balls.
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