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.

Let $Y$ be the (Tychonoff) product of an uncountable family of nontrivial Hausdorff spaces, e.g., two-point discrete spaces. $Y$ is Hausdorff because any product of Hausdorff spaces is Hausdorff. $Y$ is not metrizable because it is not first-countable. Let $X$ be a discrete space with the same cardinality as $Y$ and let $f:X\to Y$ be a surjection; then $X$ is metrizable and $f$ is continuous.
