Dense extremally disconnected subspaces

Abstract

We prove that every compact Basically Disconnected space of π \pi -weight ω 1 {\omega _1} has a dense Extremally Disconnected subspace. In Boolean algebraic terms: every σ \sigma -complete Boolean algebra B with density ω 1 {\omega _1} carries an ultrafilter which generates an ultrafilter in the completion of B. The statement that every compact Basically Disconnected space of weight c \mathfrak {c} has a dense Extremally Disconnected subspace is shown to be equivalent to CH.