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Reference: this book, page 493.

For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and $\text{Cap}(X,\epsilon)= \max \{n \, : \, X \text{ contains $n$ disjoint } \epsilon/2\text{-balls}\} $. Then we have the following

Gromov compactness theorem. Let $\mathcal{C} \subset \mathcal{M}$ (the class of all compact metric spaces) be a class of compact metric spaces. The following are equivalent:

  1. $\mathcal{C}$ is precompact, i.e. every sequence in $\mathcal{C}$ contains a subsequence which converges in $\mathcal{M}$ (in the Gromov-Hausdorff metric).
  2. There exists a function $N(\epsilon) : (0, \beta) \to (0, \infty)$ such that $\text{Cap}(X,\epsilon) \le N(\epsilon)$ for all $\epsilon \in (0, \beta)$ and $X \in \mathcal{C}$.
  3. There exists a function $N(\epsilon) : (0, \beta) \to (0, \infty)$ such that $\text{Cov}(X,\epsilon) \le N(\epsilon)$ for all $\epsilon \in (0, \beta)$ and $X \in \mathcal{C}$.

Now, let's put $X_n = \{1,n\}$. For every $\epsilon > 0$ we have that $B_{\epsilon}(1) = \{x \in X_n \, : \, |x-1| < \epsilon \} = \{1\}$ and that $B_{\epsilon}(n) = \{x \in X_n \, : \, |x-n| < \epsilon \} = \{n\}$; therefore $N(\epsilon) = 2$ for every $X_n$, but clearly $\{X_n\}_{n \in \mathbb{N}}$ doesn't admit a convergent subsequence.

It seems to me that a hypothesis is missing - i.e. a "uniform control" on the diameters of every set in $\mathcal{C}$. This is the second textbook in which I've found this (wrong?) version of the theorem (in the Gromov's original paper, for example, Gromov uses the uniform compactness).

Am I wrong?

Thank you in advance.

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  • $\begingroup$ Just to clarify: Gromov's original paper which you linked says explicitly that the "diameters are uniformly bounded" in the definition of uniform compactness for a family of compact metric space. So that resolves the issue entirely for that paper, right? (What confused me is your use of the phrase "trots out" which usually has a demeaning implication). $\endgroup$ – Lee Mosher Mar 28 '15 at 15:06
  • $\begingroup$ Yes, you are right. Indeed I've found the same hypothesis of a uniform bound for the diameters in another textbook - A Course in Metric Geometry by Burago, Burago and Ivanov, but in one more textbook (Selected Topics in Analysis in Metric Space by Ambrosio and Tilli) this hypothesis lacks again. It is a quite "funny" thing. About the expression "trots out", it is unclear, I agree. I simply meant that Gromov, in his paper, uses the hypothesis of uniform compactness. I am going to edit the OP. $\endgroup$ – gangrene Mar 28 '15 at 16:33

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