The set of maximal points of an $\boldsymbol{\omega}$-domain need not be a $\boldsymbol{G}_{\boldsymbol{\delta}}$-set

Abstract A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$ . Lawson proved that every Polish space $X$ has an $\omega$ -domain model $P$ and for such a model $P$ , $\mbox{Max}(P)$ is a $G_{\delta }$ -set of the Scott space of $P$ . Martin (2003) then asked whether it is true that for every $\omega$ -domain $Q$ , $\mbox{Max}(Q)$ is $G_{\delta }$ -set of the Scott space of $Q$ . In this paper, we give a negative answer to Martin’s long-standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for $\omega$ -algebraic domains. In addition, we also construct an $\omega$ -ideal domain $\widetilde{Q}$ for the constructed $Q$ such that their maximal point spaces are homeomorphic. Therefore, $\textrm{Max}(Q)$ is a $G_\delta$ -set of the Scott space of the new model $\widetilde{Q}$ .