On the strength of some topological lattices

TL;DR

This paper investigates the logical complexity of topological lattices such as closed, semi-linear, semi-algebraic, and convex sets using weak monadic second order logic, refining prior results and addressing an open question.

Contribution

It advances the understanding of the model theoretic strength of topological lattices and provides new insights into the algebra of convex sets.

Findings

Sharpens previous results by Grzegorczyk

Analyzes the logical strength of various topological lattices

Answers a question on the algebra of convex sets

Abstract

We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by Grzegorczyk. We also answers a question of Grzegorczyk on the 'algebra of convex sets'.