I am trying to understand quotient sets with an example of rotations in the plane. I have seen that $\mathbb R^2/SO(2)\cong \mathbb R_{>0}$. As I understand this, the quotient set is the set of orbits or equivalence classes, i.e. the set of subsets of the plane that belong to the same orbit under the action of $SO(2)$. In this case, the orbits can each be labelled by a radius $r$.
I am wondering if the quotient set $\mathbb R^2/SO(2)$ is the set of radii, which is just the non-negative real line or is it sets of subsets of all the points that lie on a given circle? More generally, is the quotient set a set of labels or a set of sets?
$$ [r_1,\dots,r_n]\qquad [[x_1,\dots,x_m],\dots[y_1,\dots,y_m]] $$
If they are distinct, how does the universal property or G-invariant maps work in this case, if there was a function that mapped points in the plane to the orbit radius $f:\mathbb R^2\rightarrow \mathbb R$, that was a $G$-invariant map and a function $\tilde f:\mathbb R^2/SO(2)\rightarrow \mathbb R$ from the quotient set.