More on countably compact, locally countable spaces

Following [5], a T ₃ space X is called good (splendid) if it is countably compact, locally countable (and ω-fair). G(κ) (resp. S(κ)) denotes the statement that a good (resp. splendid) space X with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unless ω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+{box drawings light down and left}G(ω_ω+₁); (iii) the "Chang conjecture" (ω_ω+₁),→(ω ₁, ω) implies {box drawings light down and left}S(κ) for all κ≧k≧ω_ω; (iv) if P adds ω ₁ dominating reals to V iteratively then, in [Figure not available: see fulltext.], we have G(λ^ω) for all λ.