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

A topological space has a domain model if it is homeomorphic to the maximal point space $Max(P)$ of a domain $P$. Lawson proved that every Polish space $X$ has an $\omega$-domain model $P$ and for such a model $P$, $Max(P)$ is a $G_{\delta }$-set of the Scott space of $P$. Martin (2003) then asked whether it is true that for every $\omega$-domain $Q$, $Max(Q)$ 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.