Let $f : X → Y$ be a given function, and suppose that $f^{-1}(C)$ is an open subset of $X$ whenever C is an open subset of $Y$ .
(a) Prove that f is continuous on $X$.
(b) Prove that $f^{-1}(B)$ is a closed subset of $X$ whenever B is a closed subset of $Y$
(c) If $Y = R$, and $f$ is continuous, and a $\epsilon$ $R$, what kind of set is A = {$x$ $\epsilon$ X : $f(x)$ <= a}? Justify your answer
I already solved part a, and my attempt for part b is:
$f^{−1}(B)$ = $(f^{−1}(B^c))^c$ ⋯ (1) ($E^c$ denoting the complement of $E$).
So if B is closed, then $B^c$ is open, $f^{−1}{(B^c)}$ is open and its complement is closed. This means $f^{−1}(B)$ is closed by (1).
But I'm finding trouble in solving part c. Any help please?
$f^{-1}(B)=(f^{-1}(B^{c}))^{c}$ $\cdots\, \,\,\, $ (1) ($E^{c}$ denoting the complement of $E$). So if $B$ is closed, then $B^{c}$ is open, $f^{-1}(B^{c})$ is open and its complement is closed. This means $f^{-1}(B)$ is closed by (1).