Proposition

A continuous surjective map $q: X \rightarrow Y$ is a quotient map if and only if it takes saturated open subsets to open subsets, or saturated closed subsets to closed subsets.

Solution

Suppose $q$ is a quotient map. Let $U \subset X$ be a saturated open subset. As we showed before, $U = q^{-1}(q(U))$. Since $q$ is a quotient map and $U$ is open, $q(U)$ must be open.

Let $C \subset X$ be a saturated closed subset. Then $C = q^{-1}(q(C))$. We claim that $q(C)$ is closed in $Y$. $q^{-1}(Y \setminus q(C)) = q^{-1}(Y) \setminus q^{-1}(q(C)) = X \setminus C$, which is open. Therefore, $Y \setminus q(C)$ is open, so $q(C)$ is closed.

Suppose that $q$ is a continuous surjective map that takes saturated open subsets to open subsets. We will show that $q$ is a quotient map. It suffices to show that if $q^{-1}(V)$ is open, $V$ is open in $Y$. Let $V \subset Y$ be given such that $q^{-1}(V)$ is open.

Then, $q^{-1}(V)$ is a saturated open subset. This implies that $q(q^{-1}(V))$ is open since $q$ takes a saturated open subset to an open subset. Since $q$ is surjective, $q(q^{-1}(V)) = V$. Therefore, $V$ is open in $Y$.

We have shown that $q^{-1}(V)$ is open if and only if $V$ is open. Therefore, $q$ is a quotient map.

Suppose that $q$ is a continuous surjective map that takes saturated closed subsets to closed subsets. We will show that $q$ is a quotient map. It suffices to show that if $q^{-1}(V)$ is open, $V$ is open in $Y$. Let $V \subset Y$ be given such that $q^{-1}(V)$ is open.

Then $X \setminus q^{-1}(V)$ is closed. Moreover, $X \setminus q^{-1}(V) = q^{-1}(Y) \setminus q^{-1}(V) = q^{-1}(Y \setminus V)$. Therefore, $q^{-1}(Y \setminus V)$ is a saturated closed set. This implies that $q(q^{-1}(Y \setminus V))$ is closed since $q$ takes a saturated closed subset to a closed subset. Since $q$ is surjective, $q(q^{-1}(Y \setminus V)) = Y \setminus V$. Therefore, $Y \setminus V$ is closed in $Y$, so $V$ is open in $Y$.

We have shown that $q^{-1}(V)$ is open if and only if $V$ is open. Therefore, $q$ is a quotient map.