## Quantified modal logic on the rational loine

In the topological semantics for propositional modal logic, S4 is known
to be complete wrt to the class of all topological spaces, wrt the rational line and wrt the real line. In the topological semantics for quantified modal logic, QS4 is known to be complete wrt to the class of all topological spaces, and wrt to the set of subspaces of the irrational line. The main theorem of the current paper is that QS4 is complete wrt the rational line.