σ-COMPLETE BOOLEAN ALGEBRAS AND BASICALLY DISCONNECTED COVERS

Abstract

In this paper, we show that for any $\sigma$-complete Boolean subalgebra $\mathcal M$ of $\mathcal{R}(X)$ containing $Z(X)^\#$, the Stone-space $S(\mathcal M)$ of $\mathcal M$ is a basically diconnected cover of $\beta X$ and that the subspace $\{\alpha \mid \alpha$ is a fixed $\mathcal M$-ultrafilter$\}$ of the Stone-space $S(\mathcal M)$ is the the minimal basically disconnected cover of $X$ if and only if it is a basically disconnected space and $\mathcal M\subseteq \{\Lambda_X(A) \mid A \in Z(\Lambda X)^\#\}$.