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 $\set {U_{s, n} : n \in \N}$, such that:
 * $(1): \quad \forall s \in S, n \in \N : U_{s, n + 1} \subseteq U_{s, n}$
 * $(2): \quad \forall s \in S, n \in \N : \exists m > n : \forall t \in S : U_{t, m} \cap U_{s ,m} \ne \O \implies U_{t, m} \subseteq U_{s, n}$
 * $(2): \quad \forall s \in S, n \in \N : \exists m > n : \forall t \in S : U_{t, m} \cap U_{s ,m} \ne \O \implies U_{t, m} \subseteq U_{s, n}$