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On Cantor’s Intersection Theorem in C(K) Spaces

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Abstract

We are concerned in this note with the extension of Cantor’s intersection theorem to C(K) spaces. First we prove that the general version for arbitrary closed and bounded order intervals leads to a characterization of finite dimensional C(K) spaces. On the contrary, the less restrictive version for intervals with continuous bounding functions turns out to be a characterization of injective C(K) spaces. An intersection property dealing with the Hausdorff convergence of a nested family of intervals is also proved to imply finite dimension. Finally, two particular cases related to equicontinuity and diametrical maximality are considered.

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Acknowledgements

This research was partly supported by Ministerio de Ciencia e Innovación, grant MTM2015-65825-P.

The author thanks J. M. F. Castillo, A. S. Granero and the anonymous referees for interesting suggestions and comments.

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  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, E-28049, Madrid, Spain

    J. P. Moreno

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  1. J. P. Moreno
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Moreno, J.P. On Cantor’s Intersection Theorem in C(K) Spaces. Set-Valued Var. Anal 27, 119–128 (2019). https://doi.org/10.1007/s11228-017-0424-9

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