By definition, a basis of the product topology on $$ \prod_{\alpha \in J} X_\alpha $$ consists of sets of the form $$ \prod_{\alpha \in J} U_\alpha, $$ where the $U_\alpha \neq X_\alpha$ only finitely many values of $\alpha$.
Then, could there be infinitely many sets between $U_{\alpha_1} \neq X_{\alpha_1}$ and $U_{\alpha_2} \neq X_{\alpha_2}$?
The answer depends on the cardinality of the set $J$ and it's easy to see there are no such basis for the finite $J$. Then how about the countably infinite $J$？ If there were no such basis for the countably infinite $J$, why？
If there are, then how could the metric $D(x, y)$ induce the product topology on $\mathbb{R}^\omega$ in this theorem?
Theorem. Let $\bar{d}(a, b) = \min\{|a - b|, 1\}$ be the standard bounded metric on $\mathbb{R}$. If $\mathbf{x}$ and $\mathbf{y}$ are two points of $\mathbb{R}^\omega$, define $$ D(\mathbf{x}, \mathbf{y}) = \sup \biggl\{ \frac{\bar{d}(x_i, y_i)}{i} \biggr\}. $$ Then $D$ is a metric that induces the product topology on $\mathbb{R}^\omega$.