Compact spaces, elementary submodels, and the countable chain condition

Given a space 〈 X , J 〉 in an elementary submodel M of H ( θ ) , define X M to be X ∩ M with the topology generated by { U ∩ M : U ∈ J ∩ M } . It is established, using anti-large-cardinals assumptions, that if X M is compact and its regular open algebra is isomorphic to that of a continuous image of some power of the two-point discrete space, then X = X M . Assuming CH + SCH (the Singular Cardinals Hypothesis) in addition, the result holds for any compact X M satisfying the countable chain condition.