Variations of consonance and the rational numbers

It is a result of A. Bouziad that every regular, first countable, totally imperfect space with no isolated points is not Fin-trivial. We prove that every regular totally imperfect space containing a copy of the rational numbers is not Fin-trivial in a strong sense. Our result generalizes that of Bouziad to a larger class of spaces and gives a strengthened conclusion. As a corollary we conclude that various splitting topologies on the space of continuous real-valued functions defined on a metric space need not coincide.