Let $f:X \to Y$ be a quotient map and $g: Y \to Z$, $g \circ f$ continuous. Show that $g$ is continuous.

Question

Let $f:X \to Y$ be a quotient map and $g: Y \to Z$, $g \circ f$ continuous. Show that $g$ is continuous.

Let $U$ be open in $Z$ now to show that $g$ is continuous I have to show that $g^{-1}[U]$ is open in $Y$. I have that $$g^{-1}[U]=f[(g \circ f)^{-1}[U]] = f[f^{-1}[g^{-1}[U]]]$$ and since $(g \circ f)$ is continuous the set $(g \circ f)^{-1}[U]$ is open in $X$. So I would need that $f$ is an open in order for this to work, but the quotient maps are not neccessarily open. Is the problem statement missing this info or is there another approach?

Answer 1

Let $U \subset Z$ be open.

Then $g^{-1}[U]$ is open $Y$ iff $f^{-1}[g^{-1}[U]]$ is open in $X$ as $f$ is quotient.

Now note that $f^{-1}[g^{-1}[U]] = (g \circ f)^{-1}[U]$ and draw your conclusions...