I am working on the following problem:
Let M be a set with a distance function $D$ satisfying the postulates for a metric space except that axiom $D(a,b)>0$ for $a\neq b$ is weakend to $D(a,b)\geq0$. Define $a\sim b$ to mean $D(a,b)=0$. Prove that $\sim$ is an equivalence relation. Make the set of equivalence classes into a metric space in a natural way.
So I think I have managed to show that it is an equivalence relation with reflexivity, symmetry and transitivity but the second part of the question I have no idea. What does it even mean to "make the set of equivalence classes into a metric space in a natural way"? Any help is appreciated!