Second-countability is countable product-closed

This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
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Statement

A product of countably many second-countable spaces, given the product topology, is also a second-countable space.

Related facts

The result is not true if we give the box topology to the product space.

Proof

Given: A countable family of spaces $X_{n},n\in \mathbb {N}$, with each $X_{n}$ having a countable basis $B_{n,m}$. $X$ is the product of the $X_{n}$s, given the product topology

To prove: $X$ is a second-countable space