> [!theorem] > > Let $\seq{(X_n, \rho_n)}$ be a countable family of [[Metric Space|metric spaces]], with $\rho_n$ taking values in $[0, 1]$[^1]. Let $X = \prod_{n \in \nat}X_n$, and define a metric > $ > \rho(x, y) = \sum_{n \in \nat}2^{-n}\rho_n(x_n, y_n) > $ > then $(X, \rho)$ is a metric space with the [[Product Topology|product topology]]. [^1]: Can be obtained via a transform function $t \mapsto t/(t+1)$.