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Metric Spaces

  1. Oct 24, 2011 #1
    1. The problem statement, all variables and given/known data

    [PLAIN]http://img833.imageshack.us/img833/6932/metric2.jpg [Broken]

    3. The attempt at a solution

    I've shown [itex]d_{X\times Y}[/itex] is a metric by using the fact that [itex]d_X[/itex] and [itex]d_Y[/itex] are metrics.

    What is a simpler description of [itex]d_N[/itex] with [itex]d_X[/itex] the discrete metric?

    Is it just: [tex]d_N(x,y) = \left\{ \begin{array}{lr}
    N & : x\neq y\\
    0 & : x=y
    \end{array}
    \right.[/tex]
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Oct 24, 2011 #2

    micromass

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    No. How did you get there??

    Just calculate

    [tex]d_N((x_1,x_2),(y_1,y_2))[/tex]

    and see what the possible outcomes are.
     
  4. Oct 25, 2011 #3
    Yeah I see what I assumed wrong.

    [itex]d_N(x,y)[/itex] is the number of coordidates in which x and y differ.

    How do I describe these open balls?

    The definition is: [itex]B(x,r)=\{ y\in X : d(x,y)<r \}[/itex] where [itex]x\in X[/itex] and [itex]r>0[/itex] is the radius.

    So we want:

    [itex]B((0,0),1)=\{ y\in X^2 : d((0,0),y)<1 \}[/itex] ;

    [itex]B((0,0),2)=\{ y\in X^2 : d((0,0),y)<2 \}[/itex] ;

    [itex]B((0,0),3)=\{ y\in X^2 : d((0,0),y)<3 \}[/itex] .
     
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