A proof of topological completeness for S4 in (0, 1)

The completeness of the modal logic S4 for all topological spaces as well as for the real line R, the n-dimensional Euclidean space Rn and the segment (0, 1) etc. (with □ interpreted as interior) was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure K for S4 into a subspace of the Cantor space which in turn encodes (0, 1). This provides an open and continuous map from (0, 1) onto the topological space corresponding to K. The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures.