Regular Second-Countable Space is Homeomorphic to Subspace of Hilbert Cube/Lemma 1

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $\BB$ b a countable basis for $\tau$

Let:

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

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

Then:

$\AA$ is countable

Proof

We have:

$\AA \subseteq \BB \times \BB$

where $\BB \times \BB$ is the Cartesian product of $\BB$ with itself.

From Cartesian Product of Countable Sets is Countable:

$\BB \times \BB$ is countable

From Subset of Countable Set is Countable:

$\AA$ is countable

$\blacksquare$