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Topological properties of function spaces over ordinals

  • Saak GabriyelyanEmail author
  • Jan Grebík
  • Jerzy Ka̧kol
  • Lyubomyr Zdomskyy
Original Paper

Abstract

A topological space X is said to be an Ascoli space if any compact subset \(\mathcal {K}\) of \(C_k(Y)\) is evenly continuous. This definition is motivated by the classical Ascoli theorem. We study the \(k_\mathbb {R}\)-property and the Ascoli property of \(C_{p}(\kappa )\) and \(C_k(\kappa )\) over ordinals \(\kappa \). We prove that \(C_p(\kappa )\) is always an Ascoli space, while \(C_p(\kappa )\) is a \(k_\mathbb {R}\)-space iff the cofinality of \(\kappa \) is countable. In particular, this provides the first \(C_{p}\)-example of an Ascoli space which is not a \(k_\mathbb {R}\)-space, namely \(C_p(\omega _1)\). We show that \(C_k(\kappa )\) is Ascoli iff \(\mathrm {cf}(\kappa )\) is countable iff \(C_k(\kappa )\) is metrizable.

Keywords

\(C_p(X)\) \(C_k(X)\) Ascoli \(k_\mathbb {R}\)-space Ordinal space 

Mathematics Subject Classification

54C35 54F05 46A08 54E18 

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Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  • Saak Gabriyelyan
    • 1
    Email author
  • Jan Grebík
    • 2
  • Jerzy Ka̧kol
    • 3
    • 4
  • Lyubomyr Zdomskyy
    • 5
  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic
  3. 3.A. Mickiewicz UniversityPoznańPoland
  4. 4.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic
  5. 5.Kurt Gödel Research Center for Mathematical LogicUniversity of ViennaWienAustria

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