Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 2

Theorem
Let $T = \struct {S, \tau}$ be a topological space which is regular and second-countable.

Let:
 * $\BB$ be a countable basis $\BB$ for $\tau$.

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

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

For all $\tuple{U, V} \in \AA$, let:
 * $f_{U,V} : S \to \closedint 0 1$ be a Urysohn function for $U^-$ and $S \setminus V$

Then:
 * the family of continuous mappings $\family{f_{U,V}}_{\tuple{U,V} \in \AA}$ separates points from closed sets