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This question is from Wayne Patty's Topology Section 5.2.

Consider $A$ be a compact subset of a regular space and let $U$ be an open set such that $A\subseteq U$. Prove that there is an open set $V$ such that $A \subseteq V \subseteq \overline{V} \subseteq U$.

Let $p \in A$ which implies $p \in U$. Then a result is given in the book (Theorem 5.11): A $T_1$-space $(X, \mathcal T)$ is regular if and only if for each member $p$ of $X$ and each neighbourhood $U$ of $p$, there is a neighbourhood $V$ of $p$ such that $\overline{V}\subseteq U$. So, I got $ V \subseteq \overline{V} \subseteq U$.

But I am unable to prove that $A\subseteq V \subseteq \overline{V}$. I thought that I should let $V\subseteq \overline{V} \subseteq A$ but I am not able to find a contradiction.

Can you please help with that?