Let $K\subset \mathbb{R}^N$ be a compact set. We say
$K$ is "good" if the following property holds:


Given a set of open neighborhoods $\{x\in U\_x\subset \mathbb{R}^N\}\_{x\in K}$ there exists a finite set $S\subset K$ and relatively compact open subsets $x\in \tilde{U}\_x\subset U\_x$ such that


1. $\{\tilde{U}\_x\}\_{x\in S}$ covers K,
2. For all $x,y\in S$, $\tilde{U}\_x\cap \tilde{U}\_y\neq \emptyset$ implies $\tilde{U}\_x\cap \tilde{U}\_y \cap K \neq \emptyset$.


The question is: Can a compact set be ``bad"? 


While this may sound obvious, I am worried about crazy looking cantor type compact sets.



A question on compact sets
This is an attempt to formalize Christian Remling´s idea.


Cover $K$ by finitely many balls $\{B(x,r\_x): x \in S \}$ such that $\overline{B(x,2r\_x)} \subseteq U\_x$. Let $\delta$ be the minimum of the $r\_x$'s, and for each pair $x,y \in S$, find a positive real $\lambda\_{x,y}<\delta$ such that: $$\overline{B(x,r\_x) \cap B(y,r\_y)} \cap K=\emptyset \implies \overline{B(x,r\_x+\lambda\_{x,y}) \cap B(y,r\_y+\lambda\_{x,y})} \cap K=\emptyset.$$ Let $\lambda$ be the minimum of the $\lambda\_{x,y}$'s and let $C$ be the union of all the sets $\overline{B(x,r\_x+\lambda) \cap B(y,r\_y+\lambda)}$ for which $\overline{B(x,r\_x) \cap B(y,r\_y)} \cap K=\emptyset$. We can now let $$\tilde{U}\_x= B(x,r\_x+\lambda) \setminus C$$ for each $x \in S$.


Clearly $\{\tilde{U}\_x\}\_{x\in S}$ still covers $K$ because $C \cap K=\emptyset$. Also $\tilde{U}\_x\subset U\_x$ because $\lambda < \delta$. Finally, if $\tilde{U}\_x\cap \tilde{U}\_y\neq \emptyset$, then $B(x,r\_x+\lambda) \cap B(y,r\_y+\lambda)$ is not contained in $C$. This implies that $\overline{B(x,r\_x) \cap B(y,r\_y)} \cap K \neq \emptyset$ and therefore $\tilde{U}\_x\cap \tilde{U}\_y \cap K\neq \emptyset$.


