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Ascoli’s theorem for pseudocompact spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of \(C_k(X)\) is equicontinuous, where \(C_k(X)\) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively \(\omega \)-bounded. The class of selectively \(\omega \)-bounded spaces is studied.

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Acknowledgements

The author thanks Taras Banakh for useful discussion on the name “selectively \(\omega \)-bounded”. I would like to thank the anonymous referees for useful remarks and suggestions. In particular, the question of whether the conditions (i)–(iv) in Theorem 1.2 are equivalent to the k-metrizability of \(C_k(X)\) belongs to the referee.

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  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. 653, Beer-Sheva, Israel

    Saak Gabriyelyan

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  1. Saak Gabriyelyan
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Correspondence to Saak Gabriyelyan.

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Gabriyelyan, S. Ascoli’s theorem for pseudocompact spaces. RACSAM 114, 174 (2020). https://doi.org/10.1007/s13398-020-00911-6

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