A function $f: X \to Y$ is continuous if and only if $ ^{−1} (C) $ is closed in $X $ for every closed set $C$ in $Y$. Since a mapping $f$ of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $ ^{−1} (V)$ is open in $X$ for every open set $V$ in $Y$ and since a set is closed if and only if its compliment is open,
$^{−1} (E^c)= [^{−1} (E)]^c$ for every $E⊂Y$.
Is this a correct proof of this corollary from Rudin? 

Yes, the key is that the preimage behaves nicely with all the set operations. 
