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.