It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure, then \ is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure, then $$\mathsf S4$$ S4 is the logic of any dense-in-itself metrizable space. The McKinsey–Tarski Theorem relies heavily on a metric that gives rise to the topology. We give a new and more topological proof of the theorem, utilizing Bing’s Metrization Theorem.

We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

Within ZFC, we develop a general technique to topologize trees that provides a uniform approach to topological completeness results in modal logic with respect to zero-dimensional Hausdorff spaces. Embeddings of these spaces into well-known extremally disconnected spaces then gives new completeness results for logics extending S4.2.