Compact spaces, elementary submodels, and the countable chain condition

Abstract

Given a space $〈X,\mathcal{J}〉$ in an elementary submodel $M$ of $H\left(\theta \right)$, define $X_M$ to be $X\cap M$ with the topology generated by $\{U\cap M:U\in \mathcal{J}\cap M\}$. 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 $\mathrm{CH\; +\; SCH}$ (the Singular Cardinals Hypothesis) in addition, the result holds for any compact $X_M$ satisfying the countable chain condition.