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

Theorem
Let $\left({\R, \tau_d}\right)$ denote the real number line under the Euclidean topology.

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

Let $\left({\R^\omega, d}\right)$ be the Fréchet space on $\R^\omega$., where:
 * $d \left({x, y}\right) = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \left\lvert{x_i - y_i}\right\rvert} {1 + \left\lvert{x_i - y_i}\right\rvert}$

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