Suppose I have two $C^1$ functions $F$ and $G$ which go from $\mathbb{R}^N$ to $\mathbb{R}$. Suppose at a point $x_0\in\mathbb{R}^N$, I know that $F(x_0)=0$ and $G(x_0)=0$.
Suppose further that I know the gradients of $F$ and $G$ at $x_0$ are non-zero, i.e; $\nabla F(x_0)\neq 0$ and $\nabla G(x_0)\neq 0$.
Consider the zero sets of $F$ and $G$ locally around $x_0$, $Z_F =\{x|F(x)=0\}$ and $Z_G =\{x|G(x)=0\}$. By Implicit Function Theorem, we know that $Z_F$ and $Z_G$ are continuous smooth $N-1$ dimensional graphs of functions. Further, $Z_F$ and $Z_G$ each partition the space around $x_0$ locally into 2 path-connected components (the regions above and below each graph).
My question is this: Is it true that in general $Z_F$ and $Z_G$ together partition a neighbourhood of $x_0$ into at most $4$ path-connected components? If so how do I show it and if not what is a counterexample?
Your graphs may pretty well end up like this and still be $C^1$.