Scott-product compatibility for countable meet-continuous complete lattices

Determine whether every countable meet-continuous complete lattice L satisfies the Scott-product identity σ(L × L) = σ(L) × σ(L).

Background

The paper proves the Scott-product identity for countable frames, which are meet-continuous because frames satisfy the stronger distributivity law, and for Artinian meet-continuous dcpos. It also proves that equality of the Scott topology on L × L with the ordinary product of the Scott topologies implies sobriety of the Scott space of a complete lattice.

The unresolved question asks whether countability together with meet continuity, without finite distributivity or Artinianity, is sufficient for the product identity in the complete-lattice setting. An affirmative answer would imply sobriety of the Scott space by Proposition 2.2.