Countable Infinite Product of Real Number Spaces is Homeomorphic to Fréchet Metric Space

Theorem

Let $\struct {\R, \tau_d}$ denote the real number line under the Euclidean topology.

Let $T = \struct {\R^\omega, \tau} = \ds \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ denote the countable-dimensional real Cartesian space under the product topology $\tau$.

Let $\struct {\R^\omega, d}$ be the Fréchet space on $\R^\omega$, where:

$\map d {x, y} = \ds \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$

Then the topology induced by $d$ is exactly the product topology $\tau$.

Proof

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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: $7$