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

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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 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

$\AA$ is countable

$\Box$


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 $S \setminus V$

By definition of Urysohn function:

$\forall \tuple{U,V}\in \AA : f_{U,V}$ is continuous

Lemma 2

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

$\Box$


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 evaluation mapping induced by the family $\family{f_{U,V}}_{\tuple{U, V} \in \AA}$.

That is, $f$ is the mapping defined by:

$\forall s \in S: \map f s = \family{\map {f_{U,V}} s}_{\tuple{U,V}\in \AA}$

From Topological Evaluation Mapping is Continuous:

$f$ is continuous


From Evaluation Mapping on T1 Space is Embedding if Mappings Separate Points from Closed Sets:

$f$ 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 subspace of the Hilbert cube $\struct{I^\omega, d_2}$

$\blacksquare$


Sources

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