Algebraic models for T 1-spaces

Abstract

We show that the T 1-spaces are precisely the maximal point spaces of conditionally up-complete algebraic posets with the Scott topology. Moreover, we establish an equivalence between the category of T 1-spaces with a distinguished base and a certain category of so-called camps. These are conditionally up-complete, algebraic and maximized posets in which every compact element is a meet of maximal elements, and they provide essentially unique algebraic ordered models for T 1-base spaces. A T 1-space has a damp model (a domain model that is a camp) iff it has a base not containing any free filter base. From this, it follows that 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, extending 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.