A model theory of topology

Abstract

An algebraization of the notion of topology has been proposed more than seventy years ago in a classical paper by McKinsey and Tarski. However, in McKinsey and Tarski's setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation $ \sqsubseteq $ defined by $a \sqsubseteq b$ if $a$ is contained in the topological closure of $b$. A specialization poset is a partially ordered set endowed with a further coarser preorder relation $ \sqsubseteq $. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate settings, even far removed from topology.