A model theory of topology

TL;DR

This paper develops a model-theoretic framework for topology using specialization posets and semilattices, showing their embeddings into topological spaces and revealing their relevance in diverse contexts.

Contribution

It introduces specialization posets and semilattices as algebraic structures modeling topology, with embedding theorems connecting them to classical topological spaces.

Findings

Every specialization poset can be embedded into a topological space's specialization poset.

Basic topological notions are recoverable within this weak algebraic setting.

Specialization structures appear in various unrelated fields, indicating broad applicability.

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.