Nonstandard topology on function spaces with applications to hyperspaces

Render, Hermann

doi:10.1090/S0002-9947-1993-1097169-6

Nonstandard topology on function spaces with applications to hyperspaces
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by Hermann Render PDF

Trans. Amer. Math. Soc. 336 (1993), 101-119 Request permission

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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