# Cartesian product of separable metric spaces

1. Jul 26, 2011

### Andeweld

Let $X$ be the product space of a countable family $\{X_n:n\in\mathbb{N}\}$ of separable metric spaces.
If $X$ is endowed with the product topology, we know that it is again separable. Are there other topologies for $X$ such that is separable? Is there a natural metric on $X$ such that $X$ is separable and therefore have a countable base?

The general question is under what conditions on the product space $X$ the following conclusion holds:
"For any topological base $\mathcal{B}$ in $X$, the open subsets of $X$ are countable unions of sets in $\mathcal{B}$"

Thnx

2. Jul 29, 2011

### Eynstone

Let d_n be the metric on Xn. Then d = (sum over n) {(dn/dn+1)/2^n} defines such a metric on X.
The property, however, owes more to the countability axioms. If each of a finite family of spaces has a countable basis, so must the product space in product topology.