Exercise 1.3.10 of Naber's "Topology, Geometry and Gauge fields - Foundations (2nd Edition)" reads
Show that if $f:X \rightarrow Y$ is a continuous, open mapping from $X$ onto $Y$ and $X$ is second countable then $Y$ is second countable.
It seems to me that that the statement is (trivially) false. Don't we need $f$ to be surjective as well? In that case a basis for the topology on $X$ maps to a basis for a topology on $Y$ that is the basis for the topology of $Y$. I just would like to be sure that I am not missing something.
A counter example would be taking $X$ and $Y$ to be $\mathbb{R}$ but endowing $X$ with the standard topology and $Y$ with the discrete topology. Then we can just take the constant map: $f\left(x\right)=0,\ \forall x\in\mathbb{R}$. Only the topology of the domain is second countable. Would this reasoning be correct?