continuous mapping of metrizable space

Question

Let $f$ be a continuous mapping of a metrizable space $(X,\tau)$ onto a topological space $(Y,\tau_1)$ . Is $(Y,\tau_1)$ necessarily metrizable ?

Answer 1

Let $X$ be any infinite metrisable space. Let $Y$ be the same set in the cofinite topology, and let $f$ be the identity function $f(x)=x$. $f$ is continuous as $F \subseteq Y$ is closed iff $F$ is finite or $F=Y$ and in both cases $f^{-1}[F](= F)$ is closed in $X$; it's either $X$ or it's finite and in metric spaces, finite sets are always closed.

Then $Y$ is not metrisable, as it’s not even Hausdorff. But $f$ shows that it is the continuous image of $X$.