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 $T = \struct {\R^\omega, \tau} = \displaystyle \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ 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:
 * $\map d {x, y} = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$

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