Characterization of Paracompactness in T3 Space/Lemma 8

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

Let every open cover of $T$ be even.

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

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

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

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

where:
 * $V$ is seen as a relation on $X \times X$
 * $\map V x$ denotes the image of $x$ under $V$.

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

Lemma 13

 * $\cdots$

It follows that $\AA$ is a $\sigma$-discrete refinement of $\UU$.