Choice of finite open cover

Question

Let $X$ be an open set in $\mathbb{R}^n$ and let $C, K \subset X$ be compact. Now suppose $K\cap C$ and $K\cap X\setminus C$ are both non empty. Can we then find a finite open cover of $K$ such that $K \subset \bigcup_{i=1}^{n} U_{i}$ With $U_{i} \subset C$ for $i =1,2,\dots,n-1$ and $U_{n}=X\setminus C$?

Answer 1

Doesn't seem to me to be the case. Take K the closed ball centered at 0 and of radius 1 and and C some segment that is a radius of K. Then both of the intersections are non zero,but C is of empty interior,so it can't contain any opens. Hence you can't cover K the way you want to.