Strong completeness of S4 wrt the real line

In the topological semantics for modal logic, S4 is well-known to be complete wrt the rational line and wrt the real line: these are special cases of S4's completeness wrt 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, wrt the rational line. But no similarly easy amendment is available for the real line. In another paper, we prove that S4 is, in fact, strongly complete wrt any dense-in-itself metric space, the real line being a particular case. In the current paper, we give a proof of strong completeness tailored to the particular case of the real line: we believe that it usefully clarifies matters to work through a particular and important example. We proceed in two steps: first, we show that S4 is strongly complete wrt the space of finite and infinite binary sequences, equipped with the Scott topology; and then we show that there is an interior map from the real line to this space.

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