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Math Help - Open balls in metric spaces

  1. #1
    pkr
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    Open balls in metric spaces

    d(x,y) = |x-y| / 1+|x-y|
    Describe B1(0), open ball of centre 0 and radius 1;

    so i've got (|x| / 1+|x|) < 1
    Is that enough of a description? Not sure what I could say about Br(a)

    And (X,d) a metric space, fix a point o in X
    let d1(x,y)= d(x,o) + d(o,y)
    if x ≠ y and d1(x,x)=0

    Let x ≠ 0, describe Br(x) w.r.t d1 for all possible r

    Not sure at all with that one, any hints would be appreciated.
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  2. #2
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    Quote Originally Posted by pkr View Post
    d(x,y) = |x-y| / 1+|x-y|
    Describe B1(0), open ball of centre 0 and radius 1
    Here is a question for you.
    Does there exist a point p in this metric space such that p \notin B_1 (0)?
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  3. #3
    Moo
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    Hello,
    Quote Originally Posted by pkr View Post
    d(x,y) = |x-y| / 1+|x-y|
    Describe B1(0), open ball of centre 0 and radius 1;

    so i've got (|x| / 1+|x|) < 1
    Is that enough of a description? Not sure what I could say about Br(a)
    Maybe you can try to simplify !
    1+|x|>0 for any x. So you can multiply both sides by 1+|x|, without changing the inequality :
    \frac{|x|}{1+|x|}<1
    |x|<1+|x|
    which gives 1>0, for any x. It means that the inequality is true for any x.
    So the open ball is the whole set.

    And (X,d) a metric space, fix a point o in X
    let d1(x,y)= d(x,o) + d(o,y)
    if x ≠ y and d1(x,x)=0

    Let x ≠ 0, describe Br(x) w.r.t d1 for all possible r

    Not sure at all with that one, any hints would be appreciated.
    So is B_r(x)=\{y \in X ~:~ d_1(x,y)<r\} or =\{y \in X ~:~ d(x,y)<r\} ?

    Is d the same as above ?
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  4. #4
    pkr
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    For the 1st question I assumed it to be the whole metric space but wasn't sure if it was possible, couldn't find any notes on that so many thanks.

    The second part is a different question, where (X,d) is simply any metric space.
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  5. #5
    pkr
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    I would assume that's true, but i'm confused about the whole "Let x ≠ 0"
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  6. #6
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    Here is a fix, I hope.
    Use the notation B_r (x) for a ball in the original space with metric d, and <br />
\mathbb{B}_r (x) as a ball is the new space with metric d_1.
    For 0<r\le d(x,o) then \mathbb{B}_r (x) = \{x\}.
    For r> d(x,o) then \mathbb{B}_r (x) = B_{r - d(x,o)}(o).
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