Want to know the proof of theorem 3.2 in Munkres.

Question

Theorem 3.2-2 Let $X$ be a metric space; Let Y be a subspace. a subset $A$ of $Y$ is closed in Y $iif$ it has the form $$A=C\cap Y$$ where $C$ is closed in $X$.

I can prove when $A$ of $Y$ is open in $Y$ $iif$ has the form $$A=G\cap Y$$ where $G$ is open in $X$. but I don't know how we can prove above.

Answer 1

HINT: $F$ is closed iff $G=Y\setminus F$ is open.