The set of maximal points of an ω-domain need not be a G_δ-set

2023/05/06 by Gaolin Li, Chong Shen, Li, Gaolin +7
Mathematics · #06B30 #06B35 #54A05 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN)

paper · pdf · doi:10.48550/arxiv.2305.04012

openalex publication_date 2023/05/06 · openalex created_date 2023/05/10 · openalex updated_date 2026/07/28

Abstract

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 ω-domain model P and for such a model P, Max(P) is a G_δ-set of the Scott space of P. Martin (2003) then asked whether it is true that for every ω-domain Q, Max(Q) is G_δ-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 ω-algebraic domains.

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