Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 1

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $\BB$ b a countable basis for $\tau$

Let:
 * $\AA = \set{\tuple{U,V} : U, V \in \BB : U^- \subseteq V}$

where $U^-$ denotes the closure of $U$ in $T$.

Then:
 * $\AA$ is countable

Proof
For all $U \in \BB$:
 * $\tuple{\O,U} \in \AA$
 * Pf: $\O \in \BB$
 * Pf: From Closure of Empty Set is Empty Set:
 * Pf: $\quad\O^- = \O$
 * Pf: From Empty Set is Subset of All Sets:
 * Pf: $\quad\O^- \subseteq U$
 * Pf: Result follows

From Cartesian Product of Countable Sets is Countable and Subset of Countable Set is Countable:
 * $\AA$ is countable