If $E\backslash C$ is relatively open in $E$ then $C$ is relatively closed in $E$ Given that $E\subset \mathbb{R}^n$ and $C\subset E$, prove that if $E\backslash C$ is relatively open in $E$ then $C$ is relatively closed in $E$
The best I've so far been able to arrive at is the following:
We have that, by definition, there exists an open set $A\subset \mathbb{R}^n$, such that $E\backslash C = A\cap E$. Consider $(E\cap C^c)^c = E^c\cup C=E^c\cup A^c$. Now, $E\cap (E^c \cup C)=E\cap(E^c\cup A^c) \iff E\cap (E^c \cup C)=(E\cap E^c) \cup (E^c\cap A^c)\iff E\cap C = E\cap A^c \implies C\subset A^c \cap E.$
I'm not sure if I'm on the right track. Some hints would be appreciated.

Indeed $U \setminus C = O \cap E$ for some open set $O$ of Euclidean space.
Define $F = O^c$ which is closed in Euclidean space.
Now check that $F \cap E = C$, e.g. prove two inclusions.
