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

$\{\tilde{U}_x\}_{x\in S}$ covers K,

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.

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