I'm dealing with a $2$ dimensional compact Riemannian manifold on which I consider the distance induced by the metric. I would like to know the behavior of the size of an $\epsilon$-net when $\epsilon \to 0^+$.
If $M$ is my manifold i say that $X \subset M$ is an $\epsilon$-net if for every $x \in M$ there exists a $z \in X$ s.t. $d(x,z) < \epsilon$. Define $$ N(\epsilon) = \inf \{ |X| \mid X \text{ is a } \epsilon - \text{ net } \} < + \infty $$
I'm wondering if I can say something like $$ N(\epsilon) \le C\epsilon^{-2}$$
Thank you in advance!
This is false, because if $X$ is an $\epsilon$ net by your definition then any subset of $M$ containing $X$ is also an $\epsilon$ net. So, for example, the entire surface $X=M$ satisfies your definition of an $\epsilon$ net, and that is an uncountably infinite set.
I suspect that your definition of an $\epsilon$ net is missing some important clauses.