Let $f_1, \dots, f_m \colon M \to \mathbb{R}$ be smooth functions on a manifold $M$, such that $F := (f_1, \dots , f_m) \colon M \to \mathbb{R}^m$ has constant rank $k <m $ on M.
I'm trying to show now, that it's possible to construct functions $F_1, \dots F_k \colon M \to \mathbb{R}$ s.th. they are functionally independent on an open dense subset of $M$.
For that I still need to show, that if $\{B_x\}_{x\in M}$ is an open cover of some "coordinate balls", then we find a countable family $A \subset M$, s.th. $B_\alpha \cap B_\beta = \emptyset$ and $B = \bigcup_{\alpha \in A} B_\alpha$.
For every $x \in M$ we choose and denote by $f_1^x, \dots f_k^x \in \{f_1, \dots, f_m\}$ functions, which are functionally independent in $x$. Then there exists a neighborhood $x \in U_x \subset M$, s.th. $f_1^x, \dots, f_k^x$ are functionally independent on $U_x$.
Then we find an $\epsilon_x>0$ and a "coordinate ball" $B_x (\epsilon_x) := \{y \in U_x | \left|\sum_{j=1}^k (f_j^x(y))^2 - \sum_{j=1}^k (f_j^x(x))^2\right|< \epsilon_x \}\subset U_x$ for each $x \in M$.
We define $\alpha_x(y) := \left(\sum_j^k \left((f_j^x(y))^2+(f_j^x(x))^2\right) \right)$.
Furthermore we find a smooth nonnegative function $g_x \colon \mathbb{R} \to \mathbb{R}$, s.th. $g_x(t) = 0$ for $|t| > \epsilon_x$ monotonically increasing on $[-\epsilon_x,0]$ and monotonically decreasing on $[0,\epsilon_x]$. We now define $h_x(y)=g_x\left(\alpha_x(y)\right)$.
After a short computation we see that if $g_x(\alpha_x(y))+\frac{1}{2}\alpha_x'(y) g'(\alpha_x(y)) \neq 0$, then $h_x \cdot f_1^x , \dots h_x \cdot f_k^x$ are functionally independent in $B_x(\epsilon_x)$. (If we have equality we should be able to change $g_x$ just slightly, so that we have again inequality.)
Now we have that the set $\{B_x(\epsilon_x)\}$ is an open cover of $M$. Am I now able to find a countable subset $A\subset M$ and a choice of $\epsilon_a>0$, s.th. $B_a(\epsilon_a) \cap B_b(\epsilon_b) = \emptyset$ and $B = \bigcup_{a \in A} B_a(\epsilon_a)$ is an open and dense subset in $M$?