# Equivalence classes and Induced metric

1. Mar 24, 2012

### Lily@pie

$(X,\rho)$ is a pseudometric space

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

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

Attempt:
I know that from the question,

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

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

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

Thanks

Last edited: Mar 24, 2012
2. Mar 24, 2012

### mathwonk

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

3. Mar 25, 2012

### Lily@pie

From the triangle inequality,

ρ(x,y) ≤ ρ(x,a) + ρ(a,y)

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

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