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

From Inverse of Neighborhood of Diagonal Point is Neighborhood:
 * $\forall \tuple{x, x} \in \Delta_X : R^{-1}$ is a neighborhood of $\tuple{x, x}$

From Intersection of Neighborhoods in Topological Space is Neighborhood
 * $\forall \tuple{x, x} \in \Delta_X : R \cap R^{-1}$ is a neighborhood of $\tuple{x, x}$

From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
 * $R \cap R^{-1}$ is a neighborhood of $\Delta_X$ in $\struct{X \times X, \tau_{X \times X}}$