I'm stuck... I would appreciate some help let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove there exists an $r>0$ such that $d(x,K) \leq r \rightarrow x \in U$
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Hint: Since $U$ is open and $K\subseteq U,$ what can you say for each $x\in U$ (and in particular each $x\in K$)? Don't forget that $K$ is compact, so that any open cover can be reduced to a finite subcover.