Open Cover with Closed Locally Finite Refinement is Even Cover/Lemma 4

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

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

Let $\AA$ be a closed locally finite refinement of $\UU$.

For each $A \in \AA$, let $U_A \in \UU$ such that $A \subseteq U_A$.

For each $A \in \AA$, let:
 * $V_A = \paren{U_A \times U_A} \cup \paren{\paren{X \setminus A} \times \paren{X \setminus A}}$

Let $T \times T = \struct{X \times X, \tau_{X \times X}}$ denote the product space of $T$ with itself.

Then:
 * $\forall A \in \AA : V_A$ is an open neighborhood of the diagonal $\Delta_X$ in $T \times T$

Proof
Let $A \in \AA$.

By definition of closed set:
 * $X \setminus A$ is open in $T$

By definition of product topology:
 * $U_A \times U_A, \paren{X \setminus A} \times \paren{X \setminus A}$ are open in $T \times T$

By :
 * $V_A = \paren{U_A \times U_A} \cup \paren{\paren{X \setminus A} \times \paren{X \setminus A}}$ is open in $T \times T$

Let $x \in X$.

By definition of set difference:
 * either $x \in A$ or $x \in X \setminus A$.

By definition of subset:
 * either $x \in U_A$ or $x \in X \setminus A$.

By definition of cartesian product:
 * either $\tuple{x, x} \in U_A \times U_A$ or $\tuple{x, x} \in \paren{X \setminus A} \times \paren{X \setminus A}$

By definition of set union:
 * $\tuple{x, x} \in V_A$

Since $x$ was arbitrary:
 * $\forall x \in X : \tuple{x, x} \in V_A$

By definition of diagonal:
 * $\Delta_X \subseteq V_A$

Since $A$ was arbitrary:
 * $\forall A \in \AA : V_A$ is an open neighborhood of the diagonal $\Delta_X$ in $T \times T$