Frink's Metrization Theorem

Theorem

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

Then:

$T$ is metrizable

if and only if:

for all $s \in S$ there exists a countable neighborhood basis, denoted $\set {U_{s, n} : n \in \N}$, such that:

$(1): \quad \forall s \in S, n \in \N : U_{s, n + 1} \subseteq U_{s, n}$

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

Proof

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Source of Name

This entry was named for Aline Huke Frink.

Sources

• 1970: Stephen Willard: General Topology: Chapter $7$: Metrizable Spaces: $\S23$: Metrization: Problem $23\text J$