Frink's Metrization Theorem

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Then $T$ is metrizable for all $s \in S$ there exists a countable neighborhood basis, denoted $\set{U_{s,n} : n \in \N}$:
 * (a)$\quad \forall s \in S, n \in \N : U_{s,n + 1} \subseteq U_{s,n}$
 * (b)$\quad \forall s \in S, n \in \N : \exists m > n : \forall t \in S : U_{t,m} \cap U_{s,m} \neq \O \implies U_{t,m} \subseteq U_{s,n}$