# Fréchet metric

A metric which can be placed on a countable product of metric spaces . If $(X_i,d_i)$ is a countable sequence of metric spaces with uniformly bounded metrics then the function on the product space defined by $$d((x_i),(y_i)) = \sum_i 2^{-i} d_i(x_i,y_i)$$ is a metric on the product space $\prod_i X_i$: the corresponding topology is just the product topology. If the metrics $d_i$ are not uniformly bounded then they may be replaced by equivalent bounded metrics $d_i'$ such as $\max\{d_i,1\}$ or $d_i/(1+d_i)$ in this definition. It follows that a countable product of metrizable spaces is metrizable.