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Math Help - metric spaces

  1. #1
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    metric spaces

    Define a metric d on a set X having more than one element as follows:
    For every x,y \in X, d(x,y)={1 if x =/= y, 0 if x = y}
    Let b in X. Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
    NOTE: B(b,1) is an open ball centered at b with radius 1.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Define a metric d on a set X having more than one element as follows:
    For every x,y \in X, d(x,y)={1 if x =/= y, 0 if x = y}
    Let b in X. Give an example that shows the closure of B(b,1) =/= {y in Y : d(b,y)<=1}
    NOTE: B(b,1) is an open ball centered at b with radius 1.
    By definition B(b,1) = \{ x\in X | d(x,b) < 1 \} = X - \{ b \}.
    Now B\left(b,\tfrac{1}{2}\right) = \{b\} does not intersect X - \{ b\}.
    Thus, \{b\} is not a boundary point.
    Thus, \overline{ B(b,1)} = X - \{b\}.
    Now, \{x\in X: d(x,b)\leq 1\} = X.

    And we see that X\not = X-\{b\}.
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  3. #3
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    Hello,

    If you want more information, this is called the discrete metric.

    Discrete space - Wikipedia, the free encyclopedia
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