Let $M \subset R^n$ be a $k$-dimensional manifold. I want to prove that there exist a countable union of maps $r_i: V_i \to M$ such that $M = \bigcup_{i=0}^{\infty} r(V_i)$.
We say that $r_i: V_i \to M$ is a map if it is injective, regular and smooth.
What I thought is to show that every image of map $r_i(V_i)$ has countable number of rationals, which will then prove it. I know that $r_i(V_i)$ is relatively open in $M$ and thus there is some open ball in it. Then there is a rational number in that ball. However, here I got stuck, since I can't prove that all the images of $r_i$ are disjoint, so I am not so sure.
Help would be appreciated.
Let $x\in M$ and $U_x$ the domain of a chart which contains $x$, there exists an open subset $V_x$ of $\mathbb{R}^n$ such that $V_x\cap M=U_x$, this implies that there exists a ball $B(x,r_x)$ such that $B(x,r_x)\cap M\subset U_x$, you can find an element $q_x$ with rational coefficients and a rational $l_x$ such that $x\in B(q_x,l_x)\subset B(x,r_x)$, this implies that $B(q_x,l_x)\cap M$ is a domain of a chart.