A New Proof of Completeness of S4 with Respect to the Real Line

It was proved in McKinsey and Tarski [7] that every finite wellconnected closure algebra is embedded into the closure algebra of the power set of the real line R. Pucket [10] extended this result to all finite connected closure algebras by showing that there exists an open map fromR to any finite connected topological space. We simplify his proof considerably by using the correspondence between finite topological spaces and finite quasi-ordered sets. As a consequence, we obtain that the propositional modal system S4 of Lewis is complete with respect to Boolean combinations of countable unions of convex subsets of R, which is strengthening of McKinsey and Tarski’s original result. We also obtain that the propositional modal system Grz of Grzegorczyk is complete with respect to Boolean combinations of open subsets of R. Finally, we show that McKinsey and Tarski’s result can not be extended to countable connected closure algebras by proving that no countable Alexandroff space containing an infinite ascending chain is an open image of R.