Mappings from hyperspaces and convergent sequences

AbstractFor every space X, let H(X) denote its hyperspace. A selection for X is a mapping σ: H(X) → X satisfying σ(F) ϵ F for F ϵ H (X). A point of a space X is called a κ-point if it is the limit of a nontrivial convergent sequence. In this paper we prove that if X admits a selection, then X does not include a homeomorph of the one-point compactification of an uncountable discrete space; moreover, if X is countably compact, then X is sequentially compact. A space is hyadic if it is a continuous image of H(Y) for some compact Y. Let X be hyadic. We also prove that every G8-subset of X which contains a nonisolated point of X contains a κ-point of X