View source for Fréchet metric

{{TEX|done}}{{MSC|54E35}}

A [[metric]] which can be placed on a countable product of [[metric space]]s .  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 space]]s is metrizable.


==References==
* Steen, Lynn Arthur; Seebach, J.Arthur jun. ''Counterexamples in topology'' (2nd ed.) Springer (1978) ISBN 0-387-90312-7 {{ZBL|0386.54001}}