A subset A of a compact metric M is called a $\epsilon$ net if it satisfies the following conditions 


(1)$\epsilon$ dense: the neighborhood of A is the entire M


(2)$\epsilon$ separate: $\forall x, y \in A$, $d(x,y)>\epsilon$


It is a well known fact that for any $\epsilon$, there is a fintie $\epsilon$ net. 


And I wonder whether there is an uniform bound for cardinalities of all the $\epsilon$-nets of a given compact metric space(fixed $\epsilon$). May be exist a comapcat metric space,just constructing one, who have a series of $\epsilon$ net and the cardinality of these series of $\epsilon$ net are unbounded.


I think the question is negative and should involve the Hausdorff measure, dimenson and volume, but now I am confused. It will be so nice for some people to give me a answer.



a question about $\epsilon$ net of a compact metric space.
Are you asking whether there is always an upper bound on the cardinality of an $\epsilon$-separated set of points in a compact metric space $X$? If so, the answer is yes.


Find a finite $\epsilon/2$-net $N$. Let $S$ be an $\epsilon$-separated set of points. Then every point of $S$ is in $B\_{\epsilon/2}(x)$ for some $x\in N$, and no two points of $S$ lie in the same $B\_{\epsilon/2}(x)$, so $|S|\leq |N|$.


