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Let [itex]X[/itex] be the product space of a countable family [itex]\{X_n:n\in\mathbb{N}\}[/itex] of separable metric spaces.

If [itex]X[/itex] is endowed with the product topology, we know that it is again separable. Are there other topologies for [itex]X[/itex] such that is separable? Is there a natural metric on [itex]X[/itex] such that [itex]X[/itex] is separable and therefore have a countable base?

The general question is under what conditions on the product space [itex]X[/itex] the following conclusion holds:

"For any topological base [itex]\mathcal{B}[/itex] in [itex]X[/itex], the open subsets of [itex]X[/itex] are countable unions of sets in [itex]\mathcal{B}[/itex]"

Thnx

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# Cartesian product of separable metric spaces

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