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 $\text{Max}\left(P\right)$ 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 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.