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

.

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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.

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Forgive, me for not answersing your question. It just bothered me too.