Rotations correspondence to lie algebra

Question

Let $U,V$ be non-empty, proper, connected, open subsets of the sphere $S^D(1)$ of radius 1, centered about the origin in $\mathbb{R}^D$. Is the subset $A$ of SO(D), defined by $$ A \triangleq \left\{ R: \exists x\in U y \in V, Rx=y \right\}, $$ an open subset of $SO(D)$ (of codimension $0$)?

Answer 1

Hint: Let $x\in U$, the orbit map $\rho_x:SO(D)\rightarrow S^D$ defined by $\rho_x(g)=g(x)$ is continuous, $A=\cup_{x\in U},\rho_x^{-1}(V)$ is open since $\rho_x^{-1}(V)$ is open. What is important here is the fact that $V$ is open, you maybe interested in the notion of compact open topology.