Characterization of Paracompactness in T3 Space/Lemma 20

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

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 $U$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$.

Then there exists a neighborhood $W$ of the diagonal $\Delta_X$ in $T \times T$:
 * $W$ is symmetric as a relation on $X \times X$
 * the composite relation $W \circ W$ is a subset of $U$, that is, $W \circ W \subseteq U$

Proof
Let:
 * $\VV = \set{V \in \tau : V \times V \subseteq U}$

From User:Leigh.Samphier/Topology/Neighborhood of Diagonal induces Open Cover:
 * $\VV$ is an open cover of $T$

We have, $\VV$ is even.

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

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

From Cartesian Product of Subsets:
 * $\forall x \in X : \exists V \in \VV : \map R x \times \map R x \subseteq V \times V$

From Subset Relation is Transitive:
 * $\forall x \in X : \map R x \times \map R x \subseteq U$

Let $W = R \cap R^{-1}$, where $R^{-1}$ is the inverse relation of $R$ on $X \times X$.

From Intersection of Relation with Inverse is Symmetric Relation:
 * $W$ is a symmetric relation on $X \times X$

$W$ is a Neighborhood of the Diagonal $\Delta_X$
We have:
 * $\Delta_X \subseteq R$

By definition of Definition:Symmetric Relation:
 * $\Delta_X \subseteq R^{-1}$

From Set is Subset of Intersection of Supersets:
 * $\Delta_X \subseteq W$

Let $\tuple{x, x} \in \Delta_X$

From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
 * $R$ is a neighborhood of $\tuple{x, x}$

By definition of product topology:
 * $\BB = \set {V_1 \times V_2: V_1, V_2 \in \tau}$ is a basis on $T \times T$

From User:Leigh.Samphier/Topology/Characterization of Neighborhood by Basis:
 * $\exists V_1, V_2 \in \tau : \tuple{x, x} \in V_1 \times V_2 : V_1 \times V_2 \subseteq R$

By definition of Definition:Symmetric Relation:
 * $\tuple{x, x} \in V_2 \times V_1 : V_2 \times V_1 \subseteq R^{-1}$

Let $V = V_1 \cap V_2$.

By definition of Cartesian Product:
 * $x \in V_1$ and $x \in V_2$

By definition of set intersection:
 * $x \in V$

By :
 * $V \in \tau$

By definition of product topology:
 * $V \times V \in \tau_{X \times X}$

By definition of Cartesian Product:
 * $\tuple{x, x} \in V \times V$

From Intersection is Subset:
 * $V \subseteq V_1$ and $V \subseteq V_2$

From Cartesian Product of Subsets:
 * $V \times V \subseteq V_1 \times V2$ and $V \times V \subseteq V_2 \times V1$

From Set is Subset of Intersection of Supersets:
 * $V \times V \subseteq \paren{V_1 \times V2} \cap \paren{V_2 \times V1}$

From Set Intersection Preserves Subsets:
 * $\paren{V_1 \times V2} \cap \paren{V_2 \times V1} \subseteq R \cap R^{-1} = W$

From Subset Relation is Transitive:
 * $V \times V \subseteq W$

Hence:
 * $W$ is a neighborhood of $\tuple{x, x}$

Since $\tuple{x, x}$ was arbitrary, it follows:
 * for all $\tuple{x, x} \in \Delta_X$, $W$ is a neighborhood of $\tuple{x, x}$

From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
 * $W$ is a neighborhood of the diagonal $\Delta_X$ in $T \times T$

Product of Images is Subset of $U$

 * $\forall x \in X : \map W x \times \map W x \subseteq U$

From User:Leigh.Samphier/Topology/Composition of Symmetric Relation with Itself is Union of Products of Images:
 * $W \circ W = \ds \bigcup_{x \in X} \map W x \times \map W x$

From Union of Subsets is Subset:
 * $W \circ W \subseteq U$