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 $S \setminus V$

Let $\closedint 0 1^\AA$ be the product space of countable copies of $\closedint 0 1$ indexed by $\AA$, that is:
 * $I^\AA = \ds \prod_{\tuple{U, V} \in \AA} \closedint 0 1$

Let $f: S \to \closedint 0 1^\AA$ be the continuous mapping defined by:
 * $\forall s \in S: \map f s = \family{\map {f_{U,V}} s}_{\tuple{U,V}\in \AA}$

Let $F$ be a closed set in $T$.

Let $x \in S \setminus F$.

Then:
 * $\map f x \notin f \sqbrk F$