## Strong completeness of S4 wrt the real line

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line and for the real line: these are special cases of S4's completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but strongly complete, for the rational line. But no similarly easy amendment is available for the real line. In an earlier paper, we proved a general theorem: S4 is strongly complete for any dense-in-itself metric space. Strong completeness for the real line is a special case. In the current paper, we give a proof of strong completeness tailored to the special case of the real line: the current proof is simpler and more accessible than the proof of the more general result, and involves slightly different techniques. We proceed in two steps: first, we show that S4 is strongly complete for the space of finite and infinite binary sequences, equipped with a natural topology; and then we show that there is an interior map from the real line onto this space.

Keywords: modal logic, topological semantics, strong completeness, real line.