How to prove the uniform equivalence is indeed an equivalence relation on the class of metrics on X

Question

May I ask a homework question? I'm just wandering the equivalence relation is defined on two sets while the uniformly equivalent is defined on two metrics. How can they be equal? And how to prove that? Any suggestions appreciated.

Let X be a nonempty set. Two metrics d1 and d2 on X are said to be uniformly equivalent if the identity map from (X,d1) to (X,d2) and its inverse are uniformly continuous. Prove that uniform equivalence is indeed an equivalence relation on the class of metrics on X.

Answer 1

An equivalence relation is not defined between two sets, is just defined in one set.

In this case, if we write that $d_1 \sim d_2$ for "$d_1$ is uniformly equivalent to $d_2$", then, what do you need to prove that is the following:

• $d \sim d$ for any metric $d$ in $X$;

• $d_1 \sim d_2 \Rightarrow d_2 \sim d_1$ for any two metrics $d_1$ and $d_2$ in $X$; and finally,

• $d_1 \sim d_2 \ \& \ d_2 \sim d_3 \Rightarrow d_1 \sim d_3$ for any metrics $d_1$, $d_2$ and $d_3$ in $X$.

Can you proceed from here?

Answer 2

It says "on the class of metrics on X", so your set on which the equivalence relation is defined would be $\{$d|d is a metric of X$\}$.