Compact spaces, elementary submodels, and the countable chain condition, II

Given a space 〈 X , T 〉 in an elementary submodel of H ( θ ) , define X M to be X ∩ M with the topology generated by { U ∩ M : U ∈ T ∩ M } . It is established that if X M is compact and satisfies the countable chain condition, while X is not scattered and has cardinality less than the first inaccessible cardinal, then X = X M . If the character of X M is a member of M, then “inaccessible” may be replaced by “1-extendible”.