Normality and properties related to compactness in hyperspaces

Abstract

Introduction. Let X be a regular T1 topological space and 2X the space of all closed nonempty subsets of X with the finite topology [8, Definition 1.7]. In [6] Ivanova has shown that if X is a noncompact ordinal space, then 2X is nonnormal. In this paper we give a new proof of this fact. This result is then used to show that several properties of 2X are equivalent to the compactness of X. It is not known if the normality of 2X is equivalent to the compactness of X. There are some partial results in this direction though. The paracompactness of 2x is shown to be equivalent to the compactness of X and the normality of 22X is also shown to be equivalent to the compactness of X. In the last part of the paper some properties related to the countable compactness of 2x are investigated. Notation. Because of our assumptions on X, X= { {x} :xX} is a closed subset of 2X homeomorphic to X. The set 5F,(X) = { FCX: F has at most n points is also closed. Furthermore, the space 2X is Hausdorff. For notation and further basic results on hyperspaces see [7] or [8]. In particular we use (U1, U l *, Un)= {A 2x:ACU A U; and A r Ui # 0 for all i }. If each Ui is open in X, then ( U1, * * , U,,) is open in 2X and the set of such sets in 2X forms a basis for 2X. By considering such basic open sets it is clear that the set 5F(X) of finite subsets of X is dense in 2X. We denote the cardinality of a set Z by I zi .