 Solution

A base covers the space and refines intersections. On $X^k$, products of opens form a base. The weighted metric satisfies the metric axioms and has exactly this topology; convergence is coordinatewise. For $X^{\mathbb{N}}$ the metric $d(f,g)={\sum }_{n\ge 1}{2}^{-n}\min \{1,\rho (f(n),g(n))\}$ induces the stated product topology: finitely many coordinates control a basic neighbourhood and the tail is uniformly small. Thus sequences converge exactly coordinatewise.