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Math Help - closed ball always is closed in the meteric space

  1. #1
    Junior Member
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    closed ball always is closed in the meteric space

    Q: Show that a closed ball is always closed in the meteric space?

    This is my answer. Is it true?

    Let t be any point in this complement of B(a;r). Compute the distance between the center of the closed ball and t; call it d.

    By setting epsilon = d/2, the open neighborhood (ball) centered at t with radius epsilon is disjoint from the closed ball B(a;r).

    Hence, the complement of the closed ball B(a;r) is open, thus the closed ball is a closed set.


    could any one help me in this.
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  2. #2
    Senior Member
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    Re: closed ball always is closed in the meteric space

    The basic idea is correct, except that \epsilon=d/2 is insufficient for the proof. For example, let a=0 and r=4. Then \overline{B}(a;r)=[\pm 4]. If you choose t\in[\pm4]^c=(-\infty,-4)\cup(4,\infty) then you may get something like t=6, in which case B(t;d/2)=B(6;3)=(3,9), which is not disjoint from \overline{B}(a;r)=[\pm 4].

    Instead let \epsilon=d(a,t)-r, and this will suffice for the proof.
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