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.

Many applications of digital image processing now deal with threedimensional images (the third dimension can be time or a spatial dimension). In this paper we develop a topological model for digital three space which can be useful in this context. In particular, we prove a digital, three-dimensional, analogue of the Jordan curve theorem. (The Jordan curve theorem states that a simple closed curve separates the real plane into two connected components.) Our theorem here is a digital topological formulation of the Jordan-Brouwer theorem about surfaces that separate three-dimensional space into two connected components.

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We prove that countable i-weight reflects in continuous images of weight ≤ ω 1 for all Tychonoff spaces while separability reflects in continuous images of weight ≤ ω 1 for GO spaces. If X is a GO space and all continuous images of X of weight ≤ κ + have tightness at most κ, then t(X) ≤ κ. All continuous images of weight ≤ ω 1 of a GO space X have countable pseudocharacter if and only if X is hereditarily Lindelöf. Besides, all continuous images of weight ≤ ω 1 of a linearly ordered space X have G δ-diagonal if and only if X second countable.