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

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

Then $T$ is homeomorphic to a subspace of the Hilbert cube.

Proof
From Second-Countable Space is Lindelöf:
 * $T$ is a Lindelöf space

From User:Leigh.Samphier/Topology/Regular Lindelöf Space is Normal Space:
 * $T$ is a normal space

By definition of second-countable:
 * there exists 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 $T$.

Lemma 1
From Urysohn's Lemma:
 * for all $\tuple{U, V} \in \AA$ there exists a Urysohn function $f_{U,V} : S \to \closedint 0 1$ 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 mapping defined by:
 * $\forall s \in S: \map f s = \family{\map {f_{U,V}} s}_{\tuple{U,V}\in \AA}$

By definition of Urysohn function:
 * $\forall \tuple{U,V}\in \AA : f_{U,V}$ is continuous

From Continuous Mapping to Product Space:
 * $f$ is continuous

Lemma 2
Let $F$ be a closed set in $T$.

Let $x \in S \setminus F$.

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

Lemma 3
$f$ is an embedding

By definition of embedding:
 * $T$ is homeomorphic to a subspace of $I^\AA$

From Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals:
 * $I^\AA$ is homeomorphic to the Hilbert cube $\struct{I^\omega, d_2}$

where $d_2$ is a metric

From Composite of Homeomorphisms is Homeomorphism:
 * $T$ is homeomorphic to a topological subspace of $\struct{I^\omega, d_2}$

From Metric Subspace Induces Subspace Topology:
 * $T$ is homeomorphic to a topological subspace of $\struct{I^\omega, d_2}$ which has the topology induced by the subspace metric induced by $d_2$ on $H$

By definition of metrizable topology:
 * $T$ is metrizable topology