We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P , the class P∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment {Ox : x ∈X} there is Y ⊂X with Y ∈P and {Ox : x ∈ Y } =X. We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to ...