Proof of Theorem 22.2, Munkres' Topology

Question

I think the proof does not show that $f$ is continuous when it wants to show that $f$ is a quotient map if $g$ is a quotient map. Am I correct?

Answer 1

"$f$ is continuous" was already shown, see the last sentence of the previous paragraph. This is still true, so the only thing left to show that for "$g$ quotient implies $f$ quotient" is what is claimed: surjectivity of $f$ and $f^{-1}[O]$ open in $Y$ implies $O$ open in $Z$. The other implication of continuity of $f$ is always true by the previous, because $g$ quotient implies $g$ continuous and so $f$ continuous as well. That implication was already established so can be used.