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where Ω is an ordinal (or the class of ordinals) and F is a family of (unary or finitary) operations on a topological space. The natural semantics is as follows: the modality 〈α〉 is interpreted by the αth iteration of the Cantor derivative, while the modalities 〈F 〉 and (F−1) are interpreted by the image and the preimage of the operation F . We describe logics in this language that are sound w.r.t. their natural semantics. In particular, they express certain topological properties of operations, such as injectivity, continuity, opennes, closedness, continuity and discreteness, closedness and finite rank, separate continuity and separate discreteness (this is partially based on our recent work [1]). All these logics consist of Sahlqvist formulas, hence are complete, although not always w.r.t. their natural semantics. We start with modalities 〈α〉 for derivative operators. Recall that, for X a topological space, x ∈ X is a limit point of A ⊆ X iff any neighborhood of x contains a point y ∈ A \ {x}. The Cantor derivative operator takes each A ⊆ X to its derived set A′ = {x ∈ X : x is a limit point of A}. The closure cl (A) of a set A is A ∪ A′. Iterations of the derivative are defined by A = cl (A), A = (A(α))′, and A = ⋂ β<αA (β) if α is a limit ordinal. Thus the natural semantics of the modal operator 〈α〉 is

On the modal logic of the iterated Cantor derivative and finitary operations on topological spaces