Nagata-Smirnov Metrization Theorem/Lemma 1

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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 $\ds \sum_{B \mathop \in \BB_n} \map {f_{\tuple {B, n} }^2} x$ 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 $\ds \sum_{B \mathop \in \BB_m} \map {f_{\tuple {B, m} }^2} s$ converges

The result follows.

$\blacksquare$