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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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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DOI: https://doi.org/10.1007/s13398-020-00911-6
Keywords
- \(C_k(X)\)
- Ascoli
- Sequentially Ascoli
- Selectively \(\omega \)-bounded
- Pseudocompact
- Compact-covering map