Second-countability is countable product-closed

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This article gives the statement, and possibly proof, of a topological space property satisfying a topological space metaproperty
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
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Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

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_ns, given the product topology

To prove: X is a second-countable space