Topologies for ; set-valued functions and their graphs

Author:
Louis J. Billera

Journal:
Trans. Amer. Math. Soc. **155** (1971), 137-147

MSC:
Primary 54.65

DOI:
https://doi.org/10.1090/S0002-9947-1971-0273584-0

MathSciNet review:
0273584

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Abstract: We consider the problem of topologizing , the set of all closed subsets of a topological space , in such a way as to make continuous functions from a space into precisely those functions with closed graphs. We show there is at most one topology with this property, and if is a regular space, the existence of such a topology implies that is locally compact. We then define the compact-open topology for , which has the desired property for locally compact Hausdorff . The space 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0273584-0

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