Equivalence classes and Induced metric

  1. [itex](X,\rho)[/itex] is a pseudometric space

    Define:
    x~y if and only if [itex]ρ(x,y)=0[/itex]
    (It is shown that x~y is an equivalence relation)

    Ques:
    If [itex]X^{*}[/itex] is a set of equivalence classes under this relation, then [itex]\rho(x,y)[/itex] depends only on the equivalence classes of x and y and [itex]\rho[/itex] induces a metric on [itex]X^{*}[/itex].

    Attempt:
    I know that from the question,


    [itex]X^{*}=[/itex] {[a]; [itex]a\in X[/itex]} where [itex][a]={x\in X;\rho(x,a)=0}[/itex]

    But I don't know how to go about proving that [itex]\rho(x,y)[/itex] depends only on [x] and [y]. I know i need to prove that [itex]\rho(x,y)[/itex] only depends on the all the [itex]c\in X[/itex] such that [itex]\rho(c,x)=\rho(c,y)=0[/itex].

    But I just don't know where to start...

    Thanks
     
    Last edited: Mar 24, 2012
  2. jcsd
  3. mathwonk

    mathwonk 9,732
    Science Advisor
    Homework Helper

    triangle inequality? (i.e. re read the proof that ≈ is an equivalence relation.)
     
  4. From the triangle inequality,

    ρ(x,y) ≤ ρ(x,a) + ρ(a,y)

    I know that ρ(x,a) = 0 if [itex] x \in [a] [/itex] or [itex] a \in [x] [/itex] and ρ(a,y) = 0 if [itex] y \in [a] [/itex] or [itex] a \in [y] [/itex]. And this shows that ρ(x,y) depends on [x] and [y] only?

    And how do I show it induces a metric on X*
     
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