Hyperspaces in the theory of function spaces

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1971

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Abstract

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.

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