Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space

Theorem

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

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

Proof

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

$\forall x \in M: \map f x = \sequence {\map d {x, x_i} }$

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

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

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Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $37$. Fréchet Space: $8$