Characterization of Paracompactness in T3 Space/Lemma 4

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Theorem

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

Let $\VV$ be a cover of $T$.

Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$ be an open cover of $T$.

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

Then:

$\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite

Proof

Let $A \in \AA$.

By definition of refinement:

$\exists W \in \WW : A \subseteq W$

From Subsets of Disjoint Sets are Disjoint:

$\forall V \in \VV : V \cap A \ne \O \leadsto V \cap W \ne \O$

Hence:

$\set{V \in \VV : V \cap A \ne \O} \subseteq \set{V \in \VV : V \cap W \ne \O}$

We have by hypothesis:

$\set{V \in \VV : V \cap W \ne \O}$ is finite

From Subset of Finite Set is Finite:

$\set{V \in \VV : V \cap A \ne \O}$ is finite

Since $A$ was arbitrary, it follows that:

$\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite

$\blacksquare$