A New Proof of the McKinsey–Tarski Theorem

It is a landmark theorem of McKinsey and Tarski that if we interpret modal diamond as closure (and hence modal box as interior), 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.