Doesn't a metric on a set always induce a quasi-order relation?

Question

Given that we define a metric on a set A, each element of the set has a certain distance from another element.
Let's say we choose a particular element of the set and we form the equivalence classes of the elements that have the same distance from the particular element.
To each eqivalence class will then correspond a real number. This gives us a sort of order/classification that at least provide us for a way to organize our set.
In fact, it only doesn't match the antisimmetry axiom for orderded sets, but I read somewhere that this kind of relation is called a quasi-order or pre-order relation.
I'm just approaching these subjects, so I'm not sure that this will always be the case or if my reasoning is completely mistakem

Answer 1

Yes, you are right: the relation that you defined is a preorder. And it induces an order relation in the equivalence classes.