The answers to two problems on maximal point spaces of domains

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 $G_{\delta}$-set in the Scott space $\Sigma P$, 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 \"{O}nal and Vural (2015). In this paper, we provide a new approach to Bennett and Lutzer's problem.