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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 assignmentWe 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.

We prove that, for any Tychonoff X, the space Cp(X) is Kanalytic if and only if it has a compact cover {Kp : p ∈ ω ω } such that Kp ⊂ Kq whenever p, q ∈ ω ω and p q. Applying this result we show that if Cp(X) is Kanalytic then Cp(υX) is K-analytic as well. We also establish that a space Cp(X) is K-analytic and Baire if and only if X is countable and discrete. IntroductionWe identify the space P of the irrationals with ω ω so if p, q ∈ P then p q says that p(n) q(n) for all n ∈ ω. A cover K = {K p : p ∈ P} of a space X is called P-ordered if K p ⊂ K q whenever p q. The notion of a Pordered cover appears quite naturally in K-analytic spaces; it is not difficult to show that every K-analytic space has a P-ordered compact cover. Such covers were first considered by Talagrand in [12] where he proved that, for a compact space X, the space C p (X) has a P-ordered compact cover if and only if it is K-analytic; in the same paper he constructed an example of a space which is not K-analytic but has a P-ordered compact cover.Later on, Valdivia [15] studied P-ordered covers with compact and/or countably compact elements in linear topological spaces. Cascales constructed in [4] some examples of locally convex spaces which are not Kanalytic but have P-ordered compact cover; he also proved coincidence of K-analyticity and the existence of a P-ordered compact cover in the duals of angelic locally convex spaces as well as in locally convex spaces that are quasi-complete in their Mackey topology. Another progress was achieved by Cascales and Orihuela in [7] where they proved that if a space X has a weaker

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