Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space

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Theorem

Let $M = \left({A, d}\right)$ be a metric space whose induced topology is separable.

Then $M$ is homeomorphic to a subspace of the Fréchet space $\left({\R^\omega, d}\right)$ on the countable-dimensional real Cartesian space $\R^\omega$.


Proof

Let $f: M \to \R^\omega$ be the mapping defined as:

$\forall x \in M: f \left({x}\right) = \left\langle{d \left({x, x_i}\right)}\right\rangle$

where $\left\{ {x_i}\right\}$ is a countable dense subset of $A$.

It remains to be shown that $f$ is a homeomorphism.



Sources