Prove the following are equivalent:

(a) $\displaystyle \forall V \subseteq Y$, $\displaystyle V$ is an open subset of $\displaystyle Y$ iff $\displaystyle p^{-1}(V)$ is an open subset of $\displaystyle X$

(b) $\displaystyle F$ a closed subset of $\displaystyle Y$ iff $\displaystyle p^{-1}(F)$ is a closed subset of $\displaystyle X$.