Alongside the traditional Kripke semantics, modal logic also enjoys a topological interpretation, which is becoming increasingly influential. In this paper, we present various developments related to the topological derivational semantics, based on the Cantor derivative operator. We provide several characterizations of the validity of the axioms of bounded depth. We also elucidate the topological interpretation of the axioms of directedness and connectedness—which come in different forms, all of which we examine. We then prove results of soundness and completeness for all of these logics, using a range of old and new techniques.