McKinsey-Tarski Algebras: An alternative pointfree approach to topology

TL;DR

This paper introduces McKinsey-Tarski algebras as a novel pointfree algebraic framework for topology, generalizing separation axioms and providing a duality with topological spaces.

Contribution

It develops the theory of complete interior algebras (MT-algebras), establishing a duality with topological spaces and frames, and unifies separation axioms in this algebraic setting.

Findings

MT-algebras form a faithful generalization of topological spaces

A functor from MT-algebras to frames is constructed and studied

Separation axioms are unified within the MT-algebra framework

Abstract

McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames.