McKinsey Algebras and Topological Models of S4.1

The aim of this paper is to show that every topological space gives rise to a wealth of topological models of the modal logic S4.1. The construction of these models is based on the fact that every topological space defines a Boolean closure algebra (to be called a McKinsey algebra) that neatly reflects the structure of the modal system S4.1 in that all its elements satisfy the topological interpretation of the McKinsey axiom. It is shown that the class of topological models based on McKinsey algebras contains a canonical model that can be used to prove a completeness theorem for S4.1. Finally, it is proved that the McKinsey algebra MKX of a space X endowed with an α-topology satisfies Esakia’s GRZ axiom.