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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Topologies for $ 2\sp{x}$; set-valued functions and their graphs

Author: Louis J. Billera
Journal: Trans. Amer. Math. Soc. 155 (1971), 137-147
MSC: Primary 54.65
MathSciNet review: 0273584
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Abstract: We consider the problem of topologizing $ {2^X}$, the set of all closed subsets of a topological space $ X$, in such a way as to make continuous functions from a space $ Y$ into $ {2^X}$ precisely those functions with closed graphs. We show there is at most one topology with this property, and if $ X$ is a regular space, the existence of such a topology implies that $ X$ is locally compact. We then define the compact-open topology for $ {2^X}$, which has the desired property for locally compact Hausdorff $ X$. The space $ {2^X}$ with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.

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Keywords: Set-valued function, hyperspace, closed graph, proper topology, splitting topology, conjoining topology, admissible topology, multifunction, space of subsets, function space, locally compact space
Article copyright: © Copyright 1971 American Mathematical Society