Intersection of Neighborhood of Diagonal with Inverse is Neighborhood

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

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

Let $R$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct{X \times X, \tau_{X \times X}}$.

Let $R^{-1}$ denote the inverse relation of $R$ where $R$ is viewed as a relation on $X \times X$.

Then:
 * $R \cap R^{-1}$ is a neighborhood of $\Delta_X$ in $\struct{X \times X, \tau_{X \times X}}$.

Proof
Let $W = R \cap R^{-1}$.

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

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 $T \times T$