  In this chapter we study modal logics of topological spaces in the combined
language with the derivational modality and the difference modality. We give
axiomatizations and prove completeness for the following classes: all spaces,
$T_1$-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself
separable metric space, $R^n$ $(n\ge 2)$. We also discuss the correlation
between languages with different combinations of the topological, the
derivational, the universal and the difference modality in terms of
definability.
