A question on compact sets

Question

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.

Answer 1

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 $\tilde{U}_x\cap \tilde{U}_y \cap K\neq \emptyset$.