Sigma-Locally Finite Cover has Locally Finite Refinement/Lemma 2

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Theorem

Let $T = \struct{X, \tau}$ be a topological Space.

Let $\SS = \ds \bigcup_{n = 0}^\infty \SS_n$ be a cover of $X$, where each $\SS_n$ is a set of subsets of $X$ for each $n \in \N$.

For each $n \in \N$, let:

$B_n = \ds \bigcup \SS_n$

Let:

$\BB = \set{B_n : n \in \N}$

For each $n \in \N$, let:

$C_n = B_n \setminus \ds \bigcup_{i < n} B_i$

Let:

$\CC = \set{C_n : n \in \N}$

Then:

$\CC$ is a locally finite refinement of $\BB$

Proof

Lemma 1

$\BB$ is a cover of $X$

$\Box$

$\CC$ is a Cover of $X$

Let $x \in X$.

By definition of cover of set:

$\exists n \in \N : x \in B_n$

From Well-Ordering Principle:

$\exists m \in \N : x \in B_m : \forall n < m : x \notin B_n$

By definition of set difference:

$\forall n < m : x \in B_m \setminus B_n$

By definition of set intersection:

$x \in \ds \bigcap_{i < m} \paren{B_m \setminus B_i}$

From De Morgan's Laws for Set Difference:

$\ds \bigcap_{i < m} \paren{B_m \setminus B_i} = B_m \setminus \paren{\bigcup_{i < m} B_i}$

Hence:

$x \in C_m$

It follows that $\CC$ is a cover of $X$.

$\Box$

$\CC$ is a Refinement of $\WW$

From Set Difference is Subset:

$\forall n \in \N : C_n \subseteq B_n$

It follows that $\CC$ is a refinement of $\BB$ by definition.

$\Box$

$\CC$ is Locally Finite

Let $x \in X$.

By definition of cover of set:

$\exists S \in \SS : x \in S$

By definition of union:

$\exists m \in \N : S \in \SS_m$

From Set is Subset of Union:

$V \subseteq B_m$

From Set is Subset of Union:

$\forall n > m : B_m \subseteq \ds \bigcup_{i < n} B_i$

From Set Difference with Subset is Superset of Set Difference:

$\forall n > m : C_n = B_n \setminus \ds \bigcup_{i < n} B_i \subseteq B_n \setminus B_m$

From Subset of Set Difference iff Disjoint Set:

$\forall n > m : C_n \cap B_m = \O$

From Subsets of Disjoint Sets are Disjoint:

$\forall n > m : C_n \cap V = \O$

Hence:

$\set{C_n \in \CC : C_n \cap V \ne \O} \subseteq \set{C_n \in \CC : n \le m}$

The set $\set{C_n \in \CC : n \le m}$ is finite.

From Subset of Finite Set is Finite:

$\set{C_n \in \CC : C_n \cap V \ne \O}$ is finite

It follows that $\CC$ is a locally finite refinement of $\BB$ by definition.

$\blacksquare$