Visualization of standard metric on Frechet's sequence space.

Question

Suppose $X$ is the set of all sequences in $\mathbb R$.Define a metric called Frechet metric on $X$ by $d(x,y)=\sum_{n=1}^{\infty} \frac{1}{2^n} \frac{|x_n-y_n|}{1+|x_n-y_n|}$,I want to investigate that is there any way to define such a peculiar metric.What is the idea behind this construction and how to visualize the distances and open sets in this metric space?

Answer 1

In this answer I discuss the general construction of this metric (as a weighed sum of component metrics) to see that a countable product of metrisable space (here just $\Bbb R$) is metrisable in the product topology. As you see in the proof, the idea is that openness doesn't depend on "high coordinates", as these have a small contribution to the metric, just as basic open product sets are determined by finitely many coordinates.

That it induces the product topology on $\Bbb R^{\Bbb N}$ is quite useful for convergence: a sequence $((x^{(m)}_n)_n)_m$ of points of $\Bbb R^{\Bbb N}$ converges to $(y_n)_n$ iff for all $n$, $\lim_m x^{(m)}_n = y_n$ etc. So convergence is coordinatewise, and this is ensured by the product topology and thus by this choice of metric.