A New Proof of Completeness of S4 with respect to the Real Line
Guram Bezhanishvili, Mai Gehrke
Abstract:
It was proved in McKinsey and Tarski [7] that every finite
well-connected 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 from R 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.