Hyperspaces in the theory of function spaces

In this paper certain topics involving the role played by spaces of subsets in the theory of function spaces are studied. After recalling some properties of spaces of set-valued functions a space, [cursive I[raised infinity, lowered X]], which may be considered as a direct limit of hyperspaces over a topological space X, is investigated. Then, the existence of homeomorphisms, induced by continuous selections, from Y[raised X], with the compact-open topology, into Y[raised cursive I[raised infinity, lowered X]] is demonstrated. Following this, several theorems concerning hyperspaces of function spaces are obtained, and, finally, that topology which Y[raised X] inherits as a subspace of a hyperspace of XxY is examined and compared with the compact-open topology.Mathematics, Department o