After reading about the remarkable result of Tychonoff's theorem, I've been going through some exercises to better understand the product topology. At the end of the day, this one has eluded me. Perhaps someone could shed some light on why the following is so?
Suppose I have the set $X=\{0,1\}^\mathbb{N}$ equipped with the usual product topology, and I let $\{0,1\}$ have the discrete topology. I can metrize $X$ with the metric $ \rho(\{x_n\},\{y_n\})= \begin{cases} 2^{-\inf\{n\in\mathbb{N}\mid x_n\neq y_n\}}, &\{x_n\}\neq\{y_n\}\\ 0, & \{x_n\}=\{y_n\} \end{cases} $ But why exactly does this particular metric induce the product topology on $X$?