Topology Proceedings 30 No. 1 (2006), pp. 25-38

Covering Properties and Neighborhood Assignments

Ofelia T. Alas, Vladimir V. Tkachuk and Richard G. Wilson

We continue the study originated in "Classes defined by stars and neighbourhood assignment" (to appear in Topology and its Applications), where an idea of E. van Douwen used to define D-spaces was developed. Given a topological property (or a class) P, the class P^* dual to P (with respect to neighborhood assignments) consists of spaces X such that for any neighborhood assignment {O_x:x ∈ X} there is Y ⊂ X with Y ∈ P and ∪{O_x:x ∈ Y} = X. The spaces from P^* are called dually P. We show, among other things, that a dually s-compact Tychonoff space need not be Lindelöf; this solves problems 4.1 and 4.3 from "Classes defined by stars and neighbourhood assignment." We also consider the weak dual class P' of a given class P; the elements of P' are the spaces X such that for any neighborhood assignment {O_x:x ∈ X} there is a subspace Y ⊂ X such that Y ∈ P and ∪{O_x:x ∈ Y} is dense in X. We establish that pseudocompactness is weakly self-dual in the class of Tychonoff spaces but not in the class of Hausdorff spaces; we also show that the class weakly dual to the class of compact spaces contains Tychonoff spaces which are not countably compact.

Keywords: assignment, duality, Lindelöf space, neighborhood, weak duality, weakly Lindelöf space

Mathematics Subject Classification: 54H11 54C10 22A05 54D06 (54D25 54C25)

Document formats: PDF file 217.2 Kb

Topology Proceedings, Volume 30, No. 1 (2006)
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