Characterization of Paracompactness in T3 Space/Lemma 9

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

Let every open cover of $T$ be even.

Let $\BB$ be a discrete set of subsets of $X$.

Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
 * $\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$

Proof
Let:
 * $\UU = \set{ U \in \tau : \card {\set{B \in \BB : U \cap B} } \le 1}$

Lemma 19
We have :
 * $\UU$ is an even cover.

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

By definition of even cover, there exists a neighborhood $V$ of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct {X \times X, \tau_{X \times X} }$:
 * $\set{\map V x : x \in X}$ is a refinement of $\UU$

Lemma 20
From Lemma 20, there exists an open neighborhood $W$ of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
 * $W = W^{-1}$
 * $W \circ W \subseteq V$

Lemma 21
Let $x \in X$.

By definition of refinement:
 * $\exists U \in \UU : \map V x \subseteq U$

From Corollary to Image under Subset of Relation is Subset of Image under Relation:
 * $\map {W \circ W} x \subseteq \map V x$

From Subset Relation is Transitive:
 * $\map {W \circ W} x \subseteq U$

We have:

By definition of $\UU$:
 * $\card {\set{B \in \BB : U \cap B \ne \O}} \le 1$

Hence:
 * $\card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$

Since $x$ was arbitrary, it follows that:
 * $\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$