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
We have:
 * $\AA \subseteq \BB \times \BB$

where $\BB \times \BB$ is the Cartesian product of $\BB$ with itself.

From Cartesian Product of Countable Sets is Countable:
 * $\BB \times \BB$ is countable

From Subset of Countable Set is Countable:
 * $\AA$ is countable