Quasi-metrizability of products in ZF and equivalences of CUT(fin)

It is proved in ZF that if a collection of (quasi)-metric spaces is indexed by a countable union of finite sets, then the product of this collection is (quasi)-metrizable, while it is independent of ZF that every countable product of metrizable spaces is quasi-metrizable. It is also proved that if J is a non-empty set, while a space X, consisting of at least two points, is equipped with the co-finite topology, then the product XJ is quasi-metrizable if and only if both X and J are countable unions of finite sets. Several equivalences of CUT(fin) are deduced. It is shown that, in every model of ZF+¬CUT(fin), for an uncountable set J, the space NJ can be normal (even metrizable), a Cantor cube can be simultaneously metrizable, not second-countable and non-compact.