This is my understanding of the proof. Please let me know if I am correct.
If I take any two points in a trivial topology. I can find the minimum radius such that they both are in two different open sets. Now if I union them, then there is no guarantee that a third point will not be a part of the new open ball. So the union of the two points might not be another open set. So we cannot have a metric space for this topology and this topology is not in Hausdorff space.
If a topological space is Hausdorff why is it not a sufficient condition to be metrizable?