Some Results on the Countable Compactness and Pseudocompactness of Hyperspaces

Abstract

Let X be a Hausdorff space. Let 2X denote the set of all non-empty closed subsets of X. For a subset A of X, we set 2A = {F 6 2X : F ⊆ A}. Recall that the finite topology on 2X is that topology having as a sub-basis the family {2G : G is open in X} U }2X — 2F : F is closed in X). When endowed with this topology, 2X is referred to as the hyper space of X. For the fundamental properties of hyperspaces, we refer the reader to [6; 7]. Following [6], we adopt the following notation: If A0, A1, … , An are subsets of X, we set and for all .