Characterization of Paracompactness in T3 Space/Statement 4 implies Statement 5

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

If every open cover of $T$ is even then:
 * every open cover of $T$ has an open $\sigma$-discrete refinement

Proof
Let every open cover of $T$ be even.

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

Lemma 10
By definition of $\sigma$-discrete set of subsets:
 * $\AA = \ds \bigcup_{n \in \N} \AA_n$ where $\AA_n$ is a discrete set of subsets for each $n \in \N$.

Let $X \times X$ denote the cartesian product of $X$ with itself.

Let $\tau_{X \times X}$ denote the product topology on $X \times X$.

Let $T \times T$ denote the product space $\struct {X \times X, \tau_{X \times X} }$.

Lemma 11
From Lemma 11:
 * $\forall n \in \N$ there exists a neighborhood $V_n$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
 * the set of subsets $\set{V_n \sqbrk A : A \in \AA_n}$ is discrete

For each $n \in \N$, let:
 * $\VV_n = \set{V_n \sqbrk A : A \in \AA_n}$

Let:
 * $\VV = \ds \bigcup_{n \in \N} \VV_n$

Hence $\VV$ is an open $\sigma$-discrete set of subsets by definition.

By definition of refinement:
 * $\forall A \in \AA : \exists U_A \in U : A \subseteq U_A$

For each $A \in \AA$, let:
 * $W_A = U_A \cap V_n \sqbrk A$

where $n = \min \set{m \in \N : A \in \AA_m}$

Let:
 * $\WW = \set{W_A : A \in \AA}$

Lemma 12

 * $\WW$ is an open $\sigma$-discrete refinement of $\UU$

Since $\UU$ was arbitrary, then every open cover of $T$ has an open $\sigma$-discrete refinement