Some remarks on metric spaces whose product with every Lindelöf space is Lindelöf

Let us assume that Martin's Axiom holds. We prove that if $X$ is a metrizable space whose product with every Lindelöf space is Lindelöf, then for every metric $d$ on $X,$ consistent with the topology of $X, (X,d)$ is a countable union of totally bounded subsets.