Let's define a relation $R$ on the $\mathbb{R}$ of real numbers as follows:
For any elements $x, y \in \mathbb{R}$, let's define $x R y$ to mean $x - y \in \mathbb{Q}$, the set of rational numbers.
This $R$ is indeed an equivalence relation and so partitions $\mathbb{R}$ into disjoint sets, called equivalence classes, where, for each $x \in \mathbb{R}$, we define the equivalence class $[x]$ of $x$ as
$$[x] \colon= \left\{ \ y \in \mathbb{R} \ \colon \ y R x \ \right\} = \left\{ \ y \in \mathbb{R} \ \colon \ y - x \in \mathbb{Q} \ \right\}.$$
Can we characterize (any of) these equivalence classes as a countable set of points $$ \left\{ \ p_1, p_2, p_3, \ldots \right\}?$$
Now let's form a set $S$ as follows: The set $S$ contains the smallest positive element of the equivalence classes $[ x ]$, for all $x \in \mathbb{R}$.
Then is this set $S$ bounded from above in $\mathbb{R}$? If so, then what is the supremum of $S$?
In particular, can we assume that $S$ is contained in $(0, 1)$? $(0, 1]$?
Is there a name for this particular equivalence relation in the standard mathematical literature?
Every one of the equivalence classes is countable.
I think you will have trouble defining your set $S$:
What is the smallest positive element in the equivalence class $\mathbb{Q}$ itself? If you think things through you will realize that none of the equivalence classes has a smallest positive element.
It is true that every class has many representatives in every interval. That's because you can approximate any real number arbitrarily well by rationals.
I don't know whether this relation has a standard name.