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.

Note that $E\cap C=C$.
So, following your deduction, if $E\setminus C=A\cap E$ for some open set $A$, then $C=A^c\cap E$ showing that $C$ is relatively closed in $E$.
