Is every rotation invariant set in the real plane a union of circles?

Question

Suppose we have a set $S$ in $\mathbb{R}^2$ which is fixed by every rotation about the origin. Must it be the case that $S$ is a union, possibly an empty union, of circles centered at the origin? I am counting the origin itself as a circle of radius $0$.

Answer 1

Yes, because then$$S=\bigcup_{s\in S}\{(x,y)\in\mathbb R^2\mid x^2+y^2=\lVert s\rVert^2\}.\tag1$$In fact, if $s\in S$, then it is clear that $s\in\{(x,y)\in\mathbb R^2\mid x^2+y^2=\lVert s\rVert^2\}$. Therefore,$$S\subset\bigcup_{s\in S}\{(x,y)\in\mathbb R^2\mid x^2+y^2=\lVert s\rVert^2\}.\tag2$$On the other hand, if $p\in\bigcup_{s\in S}\{(x,y)\in\mathbb R^2\mid x^2+y^2=\lVert s\rVert^2\}$, then$$p\in\{(x,y)\in\mathbb R^2\mid x^2+y^2=\lVert s\rVert^2\},$$for some $s\in S$. But then $p$ can be obtained from $s$ under a rotation around the origin. And therefore $p\in S$. It follows from this and from $(2)$ that $(1)$ holds.