Characterization of Paracompactness in T3 Space/Lemma 13

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

Let every open cover of $T$ be even.

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

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

Then:
 * there exists a sequence $\sequence{V_n}_{n \in \N}$ of neighborhoods of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} }$:
 * $V_0 = V$
 * $\forall n \in \N_{> 0} : V_n$ is symmetric as a relation on $X \times X$
 * $\forall n \in \N_{> 0}$ the composite relation $V_n \circ V_n$ is a subset of $V_{n - 1}$, that is, $V_n \circ V_n \subseteq V_{n - 1}$

Lemma 14

 * $\cdots$