User:Leigh.Samphier/Topology/Nagata-Smirnov Metrization Theorem/Lemma 1

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

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

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

For each $\tuple{B, n} \in I$, let $f_{\tuple{B, n}}:S \to \closedint 0 1$:
 * $B = \set{x \in S : \map {f_{\tuple{B, n}}} x \ne 0}$

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

Proof
Let $s \in S$ and $m \in \N$.

By definition of locally finite set of subsets:
 * $\exists U \in \tau : s \in U : \set{B \in \BB_m : B \cap U \ne \O}$ is finite

Hence:
 * $\set{B \in \BB_m : s \in B}$ is finite

It follows that:
 * $\set{\tuple{B, m} \in I : \map {f_{\tuple{B, m}}} s \ne 0}$ is finite

From Generalized Sum with Finite Non-zero Summands:
 * the generalized sum $\ds \sum_{B \in \BB_m} \map {f_{\tuple{B, m}}^2} s$ converges

The result follows.