Prove that for an open and continuous map $f:X \to Y$ between topological spaces, it is $f(\tau_X)=\tau_Y$

Question

Let $(X,\tau_x),(Y,\tau_Y)$ be topological spaces and let $f:X \to Y$ be a continuous and open function. Prove that $f(\tau_X)=\tau_Y$

This is an excercise of the problems set of my course on Introductory Topology notes. I think what I've been asked to prove is not true since if I chose $X,Y$ both equiped with the discrete topology and $Y$ having more than one element, then any constant function $f$ $(f(x)=c \in Y$ for all $x \in X)$ is continuous and open but $f(\tau_X)=\{ \emptyset, \{c\} \}\not=\tau_Y$
Is this right? Perhaps I'm missunderstanding it. Any clarification or suggestion is welcome. Thanks

Answer 1

Your counterexample is good. You could also consider the fact that $Y\notin f(\tau_X)$ unless $f$ is surjective; for instance, the inclusion of a proper open set is continuous and open.

Suppose that $f$ is surjective, besides being open and continuous. It's clear that $f(\tau_X)\subseteq \tau_Y$ (by the fact $f$ is open). If $V$ is open in $Y$, then $V=f(f^{-1}(V))$ (by surjectivity of $f$), so $V\in f(\tau_X)$ (by the fact $f$ is continuous).

Answer 2

Right. In fact, a continuous function is open if and only if $f(\tau_X) \subseteq \tau_Y$, but the converse inclusion needn't hold, in general.