Characterization of Paracompactness in T3 Space/Lemma 7

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

For each $V \in \VV$, let:
 * $U_V \in \UU : V \subseteq U_V$

Let:
 * $\UU^* = \set{V^* \cap U_V : V \in \VV}$

Then:
 * $\UU^*$ is an open locally finite refinement of $\UU$

$\UU^*$ is an Open Cover of $T$
By :
 * $\forall V^* \cap U_V \in \UU^* : V^* \cap U_V \in \tau$

From Set is Subset of Intersection of Supersets:
 * $\forall V \in \VV : V \subseteq V^* \cap U_V$

By definition of cover:
 * $\forall x \in X : \exists V \in \VV : x \in V$

By definition of subset:
 * $\forall x \in X : \exists V \in \VV : x \in V^* \cap U_V$

Hence $\UU^*$ is an open cover of $T$ by definition.

$\UU^*$ is a Refinement of $\UU$
From Intersection is Subset:
 * $\forall V \in \VV : V^* \cap U_V \subseteq U_V \in \UU$

Hence $\UU^*$ is a refinement of $\UU$.

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

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

Let:
 * $\set{A \in \VV : A \cap W \ne \O} = \set{A_1, A_2, \ldots, A_k}$ where $k \in \N$

From User:Leigh.Samphier/Topology/Subset of Cover is Cover of Subset:
 * $W \subseteq \ds \bigcup_{n = 1}^k A_n$

We have:

From Finite Union of Finite Sets is Finite, to show that $W$ intersects with only finitely many elements of $\UU^*$ it suffices to show that:
 * $\forall A \in \AA : \set{V^* \cap U_V : V \in \VV : V^* \cap U_V \cap A \ne \O}$ is finite

Let $A_0 \in \AA, V \in \VV$:
 * $V^* \cap U_V \cap A_0 \ne \O$

From Subsets of Disjoint Sets are Disjoint:
 * $V^* \cap A_0 \ne \O$

Lemma 8
From Lemma 8:
 * $V \cap A_0 \ne \O$

Lemma 4
From Lemma 4:
 * $\set{V \in \VV : V \cap A_0 \ne \O}$ is finite

Consider the surjection:
 * $f: \set{V \in \VV : V \cap A_0 \ne \O} \to \set{V^* \cap U_V : V \in \VV : V^* \cap U_V \cap A_0 \ne \O}$ defined by:
 * $\forall V \in \VV: \map f V = V^* \cap U_V$

From Cardinality of Surjection:
 * $\set{V^* \cap U_V : V \in \VV : V^* \cap U_V \cap A_0 \ne \O}$ is finite