User:Leigh.Samphier/Topology/Nagata-Smirnov Metrization Theorem/Sufficient Condition

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

Let $T$ have a basis that is $\sigma$-locally finite

Then:


 * $T$ is metrizable

Proof
Let $\BB = \ds \bigcup_{n \mathop \in \N} \BB_n$ be a $\sigma$-locally finite basis where $\BB_n$ is locally finite set of subsets for each $n \in \N$.

From T3 Space with Sigma-Locally Finite Basis is Perfectly T4 Space:
 * $T$ is a perfectly $T_4$ space