On the dissonance of some metrizable spaces

We prove that for every separable, 0-dimensional metrizable space X without isolated points, such that every compact subset of it is scattered, the cocompact topology on the hyperspace of X does not coincide with the upper Kuratowski topology—that is, X is dissonant. In particular, it follows that the rational line is dissonant, and that there exist dissonant, hereditarily Baire, separable metrizable spaces.