Example of a $T_2$ non metric topological space (if possible). I´m looking for an example of a metric topological space $X$
 and a continuous surjection $f:X\to Y$
where $Y$  is Hausdorff but not metric. 
I'm wondering because my professor and I were studying the quotient topology. We were on: $X$ metric and $X/-$ Hausdorff, then $X/-$ metric. So then we asked ourselves if there was an example like the one above, but we left it because we where gonna study other stuff. So I´m still wondering.

Here is an example where $f$ is a quotient map.  Let $X=[0,1]\times\mathbb{N}$, and let $\sim$ be the equivalence relation on $X$ given by $(x,n)\sim(y,m)$ iff either $(x,n)=(y,m)$ or $x=y=0$.  Consider the quotient space $Y=X/{\sim}$ and the quotient map $f:X\to Y$.  Then $Y$ is Hausdorff but not metrizable, because it is not first-countable.  (To prove it is not first-countable, suppose you have countably many open neighborhoods of the equivalence class $[(0,0)]\in Y$, and use a diagonalization argument to construct a new open neighborhood that does not contain any of them.)
