Let X and Y be topological spaces and f : X → Y be a continuous function. Prove that for every a ∈ Y the set $f^{−1}$({a}) is a closed set. Assume that X is connected. Characterize all continuous functions f : X → Y for which $f^{−1}$ ( { a } ) is open as well.
I used the answer of Can continuity be proven in terms of closed sets? to proof the first part (for sets instead of a single element of Y). But I cannot characterize all continuous functions.
If $Y$ is $T_1$, then all sets $\{a\}$ for $a \in Y$ are closed subsets, so $f^{-1}[\{a\}]$ is closed as the inverse image of a closed set under a continuous function.
If there is some $a_0 \in Y$ such that $f^{-1}[\{a_0\}]$ is non-empty and open, and if $X$ is connected, then $f^{-1}[\{a_0\}]$ is closed, open and non-empty, and in a connected space this means that $f^{-1}[\{a_0\}] = X$ (otherwise the set and its complement disconnect $X$!). And this in turn is equivalent to $f$ being constant (with value $a_0$). Note that then all $f^{-1}[\{a\}]$ are open, as for $a \neq a_0$ the set is just the (open) empty set. (And if all inverse images of singletons are open, one of them at least is non-empty, and the previous applies).
These are the functions $f:\>X\to Y$ which are constant on the connected components of $X$.
Proof. If $f$ is constant on the connected components of $X$ then $f^{-1}\bigl(\{a\}\bigr))$ is either empty, or a union of open components of $X$, whence open for every $a\in Y$.
Conversely: Assume that $f^{-1}\bigl(\{a\}\bigr)$ is open for every $a\in Y$. Let $\Omega$ be a component of $X$, choose a point $x_0\in \Omega$, and put $a:=f(x_0)$. The sets $$\Omega_0:=f^{-1}(\{a\})\cap\Omega$$ and $$\Omega_1=\left(\bigcup_{y\in Y\setminus\{a\}} f^{-1}\bigl(\{y\}\bigr)\right)\cap\Omega$$ are open and disjoint subsets of $\Omega$ whose union is $\Omega$. As $x_0\in\Omega_0$ it follows that $\Omega_1=\emptyset$, which then implies that $f(x)\equiv a$ on $\Omega$.
