[itex](X,\rho)[/itex] is a pseudometric space(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Equivalence classes and Induced metric

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