Scott sobriety of countable meet-continuous dcpos

Determine whether the Scott space of every countable meet-continuous dcpo is sober.

Background

The paper extends the discussion beyond complete lattices to arbitrary dcpos. Its positive result shows that Artinian meet-continuous dcpos are algebraic and Scott sober, but it leaves open the case in which countability replaces Artinianity.

A positive answer would imply a positive answer to the complete-lattice sobriety question. The paper emphasizes that the product-to-sobriety implication used for complete lattices does not directly extend to arbitrary dcpos.

References

Let $P$ be a countable meet-continuous dcpo. Must its Scott space $\Sigma P$ be sober?

— Scott topologies on products of countable complete Heyting algebras  (2609.18032 - Xu, 16 Sep 2026) in Question 2.3, Section 6 (Further questions)