Fréchet metric

2020 Mathematics Subject Classification: Primary: 54E35 [MSN][ZBL]

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.

References

• Steen, Lynn Arthur; Seebach, J.Arthur jun. Counterexamples in topology (2nd ed.) Springer (1978) ISBN 0-387-90312-7 Zbl 0386.54001