Let $Y$ be a compact metric space and $A$ be closed in $Y$ and $A^\circ=\emptyset$. Show that if $U$ is a nonempty open set in $Y$, there exist a nonempty open set $V$ such that $\overline V\subset U,\overline V\cap A=\emptyset$.
Show that there exist a nonempty open set $V$ such that $\overline V\subset U,\overline V\cap A=\emptyset$ in a compact metric space
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general-topology