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A Model Theory of Topology

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Abstract

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. 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, for a, b subsets of some topological space. 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. Specialization semilattices have the amalgamation property. 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 contexts, even far removed from topology.

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Acknowledgements

We express our gratitude to anonymous referees for a careful reading of the manuscript, for many useful comments and suggestions, in particular, suggestions for further references. The advices of the referees have led to significant improvements of the manuscript, in particular making it more concise and historically more complete.

Funding

This work has been performed under the auspices of G.N.S.A.G.A. and has been partially supported by PRIN 2012 “Logica, Modelli e Insiemi”. The author acknowledges the MIUR Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Viale della Ricerca Scientifica Specializzata, 00133, Rome, Italy

    Paolo Lipparini

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Lipparini, P. A Model Theory of Topology. Stud Logica (2024). https://doi.org/10.1007/s11225-024-10107-3

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