# 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:

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

- 1955: John L. Kelley:
*General Topology*: Chapter $4$: Embedding and Metrization - 1970: Stephen Willard:
*General Topology*: Chapter $7$: Metrizable Spaces: $\S23$: Metrization: Definition $23.1$