Scott topologies on meet-continuous domains

We study Scott products and sobriety of countable meet-continuous domains, meaning meet-continuous dcpos without any additional continuity or least-element assumption. Using the complete-lattice test-family theorem of Xu and Ji, we prove finite-product equality for those domains that are $L$-dcpos, and for the weaker class whose principal ideals have suprema of all nonempty subsets. For an arbitrary family of nonempty countable meet-continuous $L$-dcpos, we prove that the Scott topology on the order product equals the product of the factor Scott topologies if and only if only finitely many factors lack a least element. We also establish sobriety under bounded completeness and under additional common-upper-bound conditions. Assuming square-product equality, sobriety is characterized by Scott closedness of common-upper-bound sections associated with irreducible Scott-closed sets, with an equivalent sequential formulation in the countable case. Two extraction lemmas extend to meet-semilattice dcpos. The finite-product and sobriety questions for general countable meet-continuous domains remain unresolved here.