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 $X \times X$ 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 $\struct {X \times X, \tau_{X \times X} }$.

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

By definition of even cover there exists a neighborhood $V$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
 * $\forall x \in X : \exists U \in \UU : V \sqbrk x = \set {y \in X : \tuple {x, y} \in V} \subseteq U$

In what follows subsets of $X \times X$ will be treated at times as a relation on $X \times X$.

Lemma 10

 * there exists a $\sigma$-discrete refinement $\AA$ of $\UU$

Lemma 11

 * there exists an open $\sigma$-discrete cover $\VV$ of $X$ such that $\AA$ is a precise refinement of $\VV$

From [User:Leigh.Samphier/Topology/Common Refinement Condition for Open Sigma-Discrete Refinement of Open Cover]:
 * there exists an open $\sigma$-discrete cover $\WW$ of $\UU$

Since $\UU$ was arbitrary, it follows that:
 * every open cover of $T$ has an open $\sigma$-discrete refinement