Topological properties of function spaces over ordinals

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.

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Authors

Corresponding author

Correspondence to Saak Gabriyelyan.

Additional information

The second author was supported by the GACR project 15-34700L and RVO: 67985840. The third author was supported by Generalitat Valenciana, Conselleria d’Educació, Cultura i Esport, Spain, Grant PROMETEO/2013/058 and by the GAČR project 16-34860L and RVO: 67985840, and gratefully acknowledges also the financial support he received from the Kurt Goedel Research Center in Wien for his research visit in days 15.04–24.04 2016. The fourth author would like to thank the Austrian Science Fund FWF (Grant I 1209-N25) for generous support for this research. The collaboration of the second and the fourth authors was partially supported by the Czech Ministry of Education Grant 7AMB15AT035 and RVO: 67985840.

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Gabriyelyan, S., Grebík, J., Ka̧kol, J. et al. Topological properties of function spaces over ordinals. RACSAM 111, 1157–1161 (2017). https://doi.org/10.1007/s13398-016-0354-7

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Keywords

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

Mathematics Subject Classification

  • 54C35
  • 54F05
  • 46A08
  • 54E18