Complete systems, elementary submodels and the tightness of upper hyperspaces

摘要

We introduce and study some completeness properties for systems of open coverings of a given topological space. A Hausdorff space admitting a system of cardinality κ satisfying one of these properties is of type G_κ. Hence, we define several new variants of the Cech number and use elementary submodels to determine further results. We introduce M-hulls and M-networks, for M elementary submodel. As an application, we give estimates for both the tightness and the Lindelbf number of a generic upper hyperspace. Two recent results of Costantini, Hola and Vitolo on the tightness of co-compact hyperspaces follow from ours as corollaries.