Munkres Introduction to Topology: Section 17 Problem 2

If $A$ is closed in $Y$, then $Y \ - \ A$ is open in $X$, and by the definition of the subspace topology, is equal to $Y \ \bigcap \ U_x$, where $U_x$ is some general open set in $X$. By setting up the equation $Y \ - \ A \ = \ Y \ \bigcap \ U_x$ and taking the complement of both sides, we find that $A \ = \ Y \ - \ (Y \ \bigcup \ U_x)$. Through some manipulation, this can be rewritten as $A \ = \ Y \ \bigcap \ (X \ - \ U_x)$. Both $Y$ and $X \ - \ U_x$ are closed in $X$, therefore $A$ equals some closed set in $X$.