School of Mathematics and Statistics, Guilin University of Technology, Guilin 541004, ChinaScott spaces of complete Boolean algebras need not be co-sober

Abstract

In this paper, we first prove that for a complete Boolean algebra $L$, the complement graph of $L$ is a \emph{KC}-space (as a subspace) and each non-singleton compact irreducible subspace of that graph generates a non-principal $k$-irreducible compact saturated set in the Scott space of the square algebra $L\times L$. We then show that every compact sequential \emph{US}-space embeds into the complement graph of a suitable complete Boolean algebra. The embedding is built from a finite tail-constraint poset and its regular-open completion. Applying the construction to van Douwen's compact Fr\'echet anti-Hausdorff \emph{US}-space gives a complete Boolean algebra $B$ whose Scott space $\Sigma~\!\!B$ is not co-sober, thereby answering negatively a question on Scott spaces of complete Boolean algebras. The same Scott space $\Sigma~\!\!B$ is non-sober and, as a Scott space of a complete lattice, is well-filtered.