Let $A \subset X$ be closed and $U \subset A$ open in $A$. Let $V$ be any open set in $X$ with $U \subset V$.
Prove $U \cup (V \setminus A)$ is open in $X$.
Let $A \subset X$ be closed and $U \subset A$ open in $A$. Let $V$ be any open set in $X$ with $U \subset V$.
Prove $U \cup (V \setminus A)$ is open in $X$.