Proof That the Cartesian Product of a Countable Product of Second Countable Metric Spaces is Second Countable

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Theorem

Letbe a countable set of second countable metric spaces.

The cartesian productis second countable.

Proof

Since eachis second countable, a countable basisexists for the topology onfor eachA basis foris the cartesian product of the bases for theWe can write it as vector, each component of which is countable, so the vector itself is a countable quantity andis second countable.