Classes defined by stars and neighbourhood assignments

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 { O x : x ∈ X } there is Y ⊂ X with Y ∈ P and ⋃ { O x : x ∈ Y } = X . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.