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 20
The sequence $\sequence{V_n}_{n \in \N}$ is now constructed using the Principle of Recursive Definition and Zermelo's Well-Ordering Theorem.

Let:
 * $\NN = \leftset{U \subseteq X \times X : U }$ is a neighborhood of the diagonal $\Delta_X$ in $\struct {X \times X, \tau_{X \times X} } \rightset{}$

From Zermelo's Well-Ordering Theorem, let:
 * $\preccurlyeq$ be a well-ordering on $\NN$

Let $g: \NN \to \NN$ be the mapping defined by:
 * $\forall U \in \NN : \map g U = \min \leftset{ V \in \NN : V \circ V \subseteq U}$ and $V$ is symmetric $\rightset{}$

From Lemma 20:
 * $\forall U \in \NN : \leftset{ V \in \NN : V \circ V \subseteq U}$ and $V$ is symmetric $\rightset{} \ne \O$

By definition of well-ordering:
 * $g$ is well-defined

From Principle of Recursive Definition, there exists exactly one 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, $V_n \circ V_n \subseteq V_{n - 1}$