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.



Multiplicity of a subcovering in spaces of given Hausdorff dimension
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).


