Scott topologies on products of countable complete Heyting algebras

RESUMO

We prove that the Scott topology commutes with arbitrary products of countably presented frames and that every such product has a sober Scott space. In particular, this holds for arbitrary products of countable complete Heyting algebras. For a family of consonant spaces, Scott-product compatibility of their open-set lattices is equivalent to consonance of their topological sum. Countable generation does not suffice: a countably generated spatial frame can have a non-sober Scott space and fail the product identity for its square. We also show that the cardinal spectra of Scott non-sober frames and spatial frames are upward closed and, under the Continuum Hypothesis, consist of all uncountable cardinals. Finally, every Artinian $T_0$ web space is a $B$-space; if it is also a $d$-space, it carries the Scott topology of an algebraic dcpo. Consequently, every Artinian meet-continuous dcpo is algebraic. This yields a dichotomy theorem: a dcpo with a non-sober Scott space must fail meet continuity or Artinianity.