If $\mbox{rank}_{x}F\neq 0$ for all $x\in \mathbb{R}^{2}$, then there is a countable family of compact sets covering $\mathbb{R}^{2}$.

Question

Suppose that $F:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a differentiable map such that $\mbox{rank}_{x}F\neq 0$ for all $x\in \mathbb{R}^{2}$.

Can we choose a countable family of compact sets $\left\{K_{n} | n\in \mathbb{N} \right\}$ and neighbourhoods $U_{n}$ of $K_{n}$ for each $n$ with the following properties:

• the interiors of the $K_{n}$ cover $\mathbb{R}^{2}$,
• the $U_{n}$ have compact clousure,
• $\left\{U_{n} | n\in\mathbb{N}\right\}$ is locally finite,
• for each $n$ there is a pair $(i,j)$ where $i,j\in \left\{1,2\right\}$ with $\frac{\partial F_{i}}{\partial x_{j}}\neq 0$ for each $a\in K_{n}$. ?