The metrization problem:
In fact, there are sufficient criteria on the topology that assure the existence
of such a metric even if this is not explicitly given. An example of an existence theorem
of this kind is due to Urysohn (Kelley 1955, p. 125), who proved that a regular T1-space
whose topology has a countable basis is metrizable.
Conversely, a metrizable space is always and regular, but the condition on the basis
has to be weakened since in general, it is only true that the topology has a basis which
is formed by countably many locally finite families of open sets.
Urysohn's Metrization Theorem.
For every topological T1-space X, the following conditions are equivalent.
Urysohn-Lemma: In a T4-space, any pair X, X' of disjoint closed subsets can be separated by
a function into the real numbers: there is such a function f with f(X) = 0, f(X') = 1.
T1 means: for any pair x,y of points, there exists an open set U with x in U, y not in U
(and also ...) - this means: one element subsets are closed.
T4 means: Disjoint closed subsets can be separated by disjoint open subsets.
T1 & T4 = "normal" (normal spaces are Hausdorff).
Separability condition, dimension condition.