Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras

Abstract:

Under ${\text {MA}} {\text { + }} {{\text {2}}^\omega } = {\omega _2}$ there is a (compact) strongly zero-dimensional F-space of weight ${2^\omega }$ which cannot be embedded in any basically disconnected space. Dually, under ${\text {MA}} + {2^\omega } = {\omega _2}$ there is a weakly countably complete (or almost $\sigma$-complete, or countable separation property) Boolean algebra of cardinality ${2^\omega }$ which is not a homomorphic image of any countably complete Boolean algebra. The key to our construction is the observation that if X is a subspace of a basically disconnected space and $\beta \omega \subseteq X$ then $\beta \omega$ is a retract of X. Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto $\mathcal {P}(\omega )$, the field of subsets of w, then there is an embedding $e: \mathcal {P}(\omega ) \to B$ such that $h \circ e = {\text {i}}{{\text {d}}_{\mathcal {P}(\omega )}}$.