Spaces with a star-countable k-network, and related results

Recall that a collection P of subsets of a space is star-countable if each element of P meets at most countably many other elements of P. We characterize the closed image of a locally separable metric space as a Fréchet space with a star-countable k-network, and give a characterization for the product of k-spaces with a star-countable k-network to be a k-space.