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Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all balls have the same radius, but it might be unnecessary.

Is it true that one can choose a subcovering such that every point of $K$ is covered by at most $N$ balls, where $N$ depends on $n$ only?

This question might be rather trivial to experts; I have very little experience with the subject.

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The answer is no.

Hausdorff dimension does not reflect any global geometry. Say you can construct a metric graph which approximates any compact length-metric space (as well as a finite metric space which approximates any compact metric space).

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