Let $X$ be a topological space with no special properties. My book claims the following in an exercise:
$A \subset X$ is closed.
$U \subset A$ is open (with respect to the induced topology on $A$).
$V \subset X$ is open with $U \subset V$.
Show: $U \cup (V - A)$ is open in $X$.
I can see why this would be true in the presence of some separation axioms, but the book asks me to prove it in general.
Many thanks for your help.