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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 cardinality cardinalities of all the $\epsilon$-nets(fixed \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$net are unbounded. I think the question is negative and should involve the Hausdorff measure, dimenson and volume, but now I am confused. It will be so nice for some people to give me a answer. 2 added 25 characters in body A subset A of a compact metric M is called a$\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$net. And I wonder whether there is an unique uniform bound for the number cardinality of all the$\epsilon$nets. \epsilon$-nets(fixed $\epsilon$). May be exist a comapcat metric space,just constructing one, who have a series of $\epsilon$ net and the number 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.

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# a question about $\epsilon$ net of a compact metric space.

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 unique bound for the number of all the $\epsilon$ nets. May be exist a comapcat metric space,just constructing one, who have a series of $\epsilon$ net and the number 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.