# Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space

Jump to navigation
Jump to search

## 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 37: \ 8$