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

Let $I = \set{\tuple{B, n} : B \in \BB, B \in \BB_n}$.

By definition of perfectly $T_4$ space:
 * $\forall \tuple{B, n} \in I : \exists f_{\tuple{B, n}} : S \to \closedint 0 1 : B = \set{x \in S : \map {f_{\tuple{B, n}}} x \ne 0}$

Let $\alpha$ be the cardinality of $I$.

Let $H^\alpha$ be the generalized Hilbert sequence space of weight $\alpha$.

Lemma 1
For all $x \in S$ and $n \in \N$:
 * the generalized sum $\ds \sum_{B \in \BB_n} \map {f_{\tuple{B, n}}^2} x$ converges

Let $g_n : S \to \closedint 0 1$ be the mapping defined by:
 * $\map {g_n} x$ is the limit of the generalized sum $\ds \sum_{B \in \BB_n} \map {f_{\tuple{B, n}}^2} x$

From Lemma 1:
 * $\forall n \in \N : g_n : S \to \closedint 0 1$ is well-defined

Lemma 2
For all $x \in S$:
 * the generalized sum $\ds \sum_{\tuple{B, n} \in I} \sqbrk{\dfrac 1 {\paren{\sqrt 2}^n} \dfrac {\map {f_{\tuple{b, n}}} x} {\sqrt {1 + \map {g_n} x}}}^2$ converges

Let $F : S \to H^\alpha$ be the mapping defined by:


 * $\ds \map F x = \family{\dfrac 1 {\paren{\sqrt 2}^n} \dfrac {\map {f_{\tuple{b, n}}} x} {\sqrt {1 + \map {g_n} x}}}_{\tuple{B, n} \in I}$

From Lemma 2:
 * $F : S \to H^\alpha$ is well-defined

$F$ is a closed mapping
From :
 * $S$ is homeomorphic to a subset $X$ of $H^\alpha$

From :
 * $X$ is a metric space

From :
 * $T$ is metrizable