Let $A\subset \mathbb{R}^{3}$ intersection of open ball centered at origin and the cone $C=\{(x,y,z)\in\mathbb{R}^{3}; x^{2}+y^{2}=z^{2},z\geq 0\}$. Prove there is not immersion $F:U\subset\mathbb{R}^{2}\to \mathbb{R}^{3}$ where $U\subset \mathbb{R}^{2}$ such that $F(U)=A$.
My idea is that if there existis $F:U\subset\mathbb{R}^{2}\to \mathbb{R}^{3}$ immersion then if $p \in U$ there is a neighboorhood $V$ of $p$ and $F|_{V}:V\to \mathbb{R}^{3}$ is an embedding then $F(V)$ is an immersed submanifold of $\mathbb{R}^{3}$. Am I going the right way?
Any tips are appreciated!