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Nonstandard topology on function spaces with applications to hyperspaces


Author: Hermann Render
Journal: Trans. Amer. Math. Soc. 336 (1993), 101-119
MSC: Primary 54J05; Secondary 03H05, 54B20, 54C35, 54D60
DOI: https://doi.org/10.1090/S0002-9947-1993-1097169-6
MathSciNet review: 1097169
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Abstract: In this paper the techniques of Nonstandard Analysis are used to study topologies on the set of all continuous functions. We obtain nonstandard characterizations for conjoining and splitting topologies and we give a complete description of the monads of the compact-open topology which leads to very elegant and simple proofs of some important results. For example we prove a generalized Ascoli Theorem where the image space is only Hausdorff or regular. Then we apply our results to the hyperspace and solve questions of Arens and Dugundji, Wattenberg and Topsøe. Finally we discuss real compact spaces and the continuity of the diagonal function.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1097169-6
Keywords: Hyperspaces, convergence topology, Ascoli Theorem, diagonal function, exponential map, real compactness
Article copyright: © Copyright 1993 American Mathematical Society

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