Characterization of Paracompactness in T3 Space/Lemma 6

Theorem
Let $T = \struct{X, \tau}$ be a $T_3$ Space.

Let $\UU$ be an open cover of $T$.

Let $\VV$ be a closed locally finite refinement of $\UU$.

For all $x \in X$, let:
 * $W_x \in \tau: x \in W_x$ and $\set{V \in \VV : V \cap W \ne \O}$ is finite

Let $\WW = \set{W_x : x \in X}$ be an open cover of $T$.

Let $\AA$ be a closed locally finite refinement of $\WW$.

For each $V \in \VV$, let:
 * $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O}$

Let $\VV^* = \set{V^* : V \in \VV}$.

Then:
 * $\VV^*$ is an open locally finite cover of $T$

$\VV^*$ is a Set of Open Subsets
Let $V^* \in \VV^*$ for some $V \in \VV$.

Let $\AA_V = \set{A \in \AA | A \cap V = \O}$.

By definition of subset:
 * $\AA_V \subseteq \AA$

From Subset of Locally Finite Set of Subsets is Locally Finite:
 * $\AA_V$ is closed locally finite

From Union of Closed Locally Finite Set of Subsets is Closed:
 * $\bigcup \set{A \in \AA | A \cap V = \O}$ is closed in $T$

By definition of closed set:
 * $V^* = X \setminus \ds \bigcup \set{A \in \AA | A \cap V = \O} \in \tau$

$\VV^*$ is a Cover
Let $x \in X$.

By definiton of a cover:
 * $\exists V \in \VV : x \in V$

From Lemma 4:
 * $V \subseteq V^*$

By definition of subset:
 * $x \in V^*$

Since $x$ was arbitrary, it follows that $\VV^*$ is a cover by definition.

$\VV^*$ is Locally Finite
Let $x \in X$.

By definition of locally finite:
 * $\exists U \in \tau : x \in U : \set{A \in \AA : A \cap U \ne \O}$ is finite.

Let $\set{A \in \AA : A \cap U \ne \O} = \set{A_1, A_2, \ldots, A_n}$ for some $n \in \N$.

From User:Leigh.Samphier/Topology/Subset of Cover is Cover of Subset:
 * $U \subseteq \ds \bigcup \set{A_1, A_2, \ldots, A_n}$

Let:
 * $V^* \in \VV^* : V^* \cap U \ne \O$

We have:

Hence:
 * $\exists i \in \set{1, 2, \ldots, n} : V^* \cap A_i \ne \O$

From Lemma 7:
 * $V \cap A_i \ne \O$

Hence:
 * $\set{V^* \in \VV^* : V^* \cap U} \subseteq \set{V^* \in \VV^* : \exists 1 \le i \le n : V \cap A_i \ne \O}$

For each $1 \le i \le n$:
 * $\exists W_i \in \WW : A_i \subseteq W_i$

By definition of $\WW$:
 * $\forall 1 \le i \le n : \set{V \in \VV : V \cap W_i \ne \O}$ is finite

Hence:
 * $\forall 1 \le i \le n : \set{V \in \VV : V \cap A_i \ne \O}$ is finite

From Union of Finite Sets is Finite:
 * $\set{V^* \in \VV^* : \exists 1 \le i \le n : V \cap A_i \ne \O}$ is finite

From Subset of Finite Set is Finite:
 * $\set{V^* \in \VV^* : V^* \cap U}$ is finite

Since $x$ was arbitrary, it follows that $\VV^*$ is locally finite.