The answers to two problems on maximal point spaces of domains
Xiaoyong Xi, Chong Shen, Dongsheng Zhao
TL;DR
The paper addresses two core questions in domain theory: whether the maximal point space $Max(P)$ of an ideal domain need not be a $G_{\delta}$-set, and whether domain-representability is preserved under products. It provides a negative answer to the first by constructing an ideal domain $L$ with $Max(L)$ not a $G_{\delta}$-set, while showing that a slight modification $\widehat{L}$ can yield a $G_{\delta}$-set, illustrating non-absolute behavior. For the second question, the authors present a direct constructive proof that if $X\times Y$ is domain-representable, then both $X$ and $Y$ are domain-representable, via a new poset-based construction that derives domain models of factor spaces from a product model. This yields a fresh approach to Bennett and Lutzer’s product problem and contributes new techniques for constructing factor-space domain models, while also raising open questions about characterizations of $Max(M)$ and extensions to Scott-domain models.
Abstract
A topological space is domain-representable (or, has a domain model) if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ (with the relative Scott topology). We first construct an example to show that the set of maximal points of an ideal domain $P$ need not be a $G_δ$-set in the Scott space $ΣP$, thereby answering an open problem from Martin (2003). In addition, Bennett and Lutzer (2009) asked whether $X$ and $Y$ are domain-representable if their product space $X \times Y$ is domain-representable. This problem was first solved by Önal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem.