Algebraic models for \(T_1\)-spaces
The author contributes to the theory of computational models of spaces. He proves that the \(T_1\)-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. He also establishes an equivalence between the category of \(T_1\)-spaces with a distinguished base and a certain category of camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements. They provide essentially unique algebraic ordered models for \(T_1\)-base spaces. A \(T_1\)-space has a damp model (that is, a domain model that is a camp) if and only if it has a base not containing any free filter base. Consequently all completely metrizable spaces and, more generally, all complete Aronszajn spaces have damp models. Moreover, damp models also exist for all Stone spaces. The latter representation gives rise to an equivalence and a duality for so-called Stone base spaces, which extends the classical Stone duality. Furthermore, it yields a purely order-theoretical description of clopen bases for Stone spaces and, algebraically, of finitary meet bases of Boolean lattices in terms of maximal ideals.
scientific article; zbMATH DE number 5888199
scientific article