I was finally able to find this on PlanetMath, (Source taken from this website below).
A topological space is said to be metrizable if there is a metric such that the topology induced by is .
This is what bothers me,
induced as in graph theory.
By the problem is that the topology can contain more than two element subsets of in that case, how can we view as a graph!
Furthermore, then for certainly the topology on does not contain two element subsets. That means that is not a graph and the entire concept of makes no sense.
Forgive, me for not answersing your question. It just bothered me too.