
#1
Mar2412, 12:53 PM

P: 109

[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 



#2
Mar2412, 01:46 PM

Sci Advisor
HW Helper
P: 9,428

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




#3
Mar2512, 03:13 AM

P: 109

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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