Completeness number of families of subsets of convergence spaces

Compactoid and compact families generalize both convergent filters and compact sets. This concept turned out to be useful in various quests, like Scott topologies, triquotient maps and extensions of the Choquet active boundary theorem. The completeness number of a family in a convergence space is the least cardinality of collections of covers for which the family becomes complete. 0-completeness amounts to compactness, finite completeness to relative local compactness and countable completeness to Cech completeness. Countably conditional countable completeness amounts to pseudocompleteness of Oxtoby. Conversely, each completeness class of families can be represented as a class of conditionally compactoid families. In this framework, the theorem of Tikhonov for compactoid filters becomes a special case of the theorem on the completeness number of products. A characterization of completeness in terms of non-adherent filters not only provides a unified language for convergence and completeness, but also clarifies preservation mechanisms of completeness number under various operations. (C) 2015 Elsevier B.V. All rights reserved.