Countable Product of Second-Countable Spaces is Second-Countable

Theorem
Let $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$ be a countable set of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$.

Let each of $\left({X_\alpha, \vartheta_\alpha}\right)$ be second-countable.

Then $\left({X, \vartheta}\right)$ is also second-countable.