Skip to main content
SpringerLink
Log in

The Selectively Pseudocompact-Open Topology on C(X)

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

This paper studies the selectively pseudocompact-open topology on the set of the real-valued continuous functions on a Tychonoff space, and compares this topology with the \(\alpha \)-open topology and the topology of uniform convergence on the elements of \(\alpha \) where \(\alpha \) is one of the following: \(\{X\}\), the finite subsets of X, the compact subsets of X, the countably compact subsets of X, and the pseudocompact subsets of X. Moreover, it also analyzes the almost cc-\(\omega \)-bounded spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alas, O.T., Martínez-Cadena, J.A., Wilson, R.G.: On sequential and selective feeble compactness. Houston J. Math. 4(3), 1063–1079 (2018)

    MathSciNet  MATH  Google Scholar 

  2. Angoa, J., Ortiz-Castillo, Y.F., Tamariz-Mascarúa, Á.: Ultrafilters and properties related to compactness. Topol. Proc. 43, 183–200 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Bruguera, M., Tkachenko, M.: The three space problem in topological groups. Topol. Appl. 153(13), 2278–2302 (2006)

    Article  MathSciNet  Google Scholar 

  4. Dorantes-Aldama, A., Shakhmatov, D.: Selective sequential pseudocompactness. Topol. Appl. 222, 53–69 (2017)

    Article  MathSciNet  Google Scholar 

  5. Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  6. García-Ferreira, S., Ortíz-Castillo, Y.F.: Strong pseudocompact properties. Comment. Math. Univ. Carolinae 55(1), 101–109 (2014)

    MathSciNet  MATH  Google Scholar 

  7. García-Ferreira, S., Tomita, A.H.: A pseudocompact group which is not strongly pseudocompact. Topol. Appl. 192, 138–144 (2015)

    Article  MathSciNet  Google Scholar 

  8. Kundu, S., Garg, P.: The pseudocompact-open topology on C(X). Topol. Proc. 30, 279–299 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Kundu, S., Garg, P.: Countability Properties of the Pseudocompact-Open Topology on C(X): A Comparative Study. Rend. Istit. Mat. Univ. Trieste 39, 421–444 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Kundu, S., Garg, P.: Completeness properties of the pseudocompact-open topology on C(X). Math. Slovaca. 58(3), 325–338 (2008)

    Article  MathSciNet  Google Scholar 

  11. Kundu, S., McCoy, R.A.: Topologies between compact and uniform convergence on function spaces. Int. J. Math. Math. Sci. 16(1), 101–109 (1993)

    Article  MathSciNet  Google Scholar 

  12. Kundu, S., McCoy, R. A., Raha, A. B.: Topologies between compact and uniform convergence on function spaces, II, Real Anl. Exchange 18, no. 1, 176–189 (1992/93)

  13. Kundu, S., Raha, A.B.: The bounded-open topology and its relatives. Rend. Istit. Mat. Univ. Trieste 27, 61–77 (1995)

    MathSciNet  MATH  Google Scholar 

  14. Martínez-Cadena, J. A., Sanchis, M.: Some aspects on generalizations of compactness in (para)topological groups, Topol. Algebra Appl., preprint

  15. McCoy, R.A., Ntantu, I.: Completeness properties of function spaces. Topol. Appl. 22, 191–206 (1986)

    Article  MathSciNet  Google Scholar 

  16. McCoy, R. A., Ntantu, I.: Topological Properties of Spaces of Continuous Functions, Lecture Notes in Math., 1315 (Springer, Berlin) (1988)

  17. Mrówka, S.: On complete regular spaces. Fund. Math. 41, 105–106 (1954)

    Article  MathSciNet  Google Scholar 

  18. Nokhrin, S.E., Osipov, A.V.: On the coincidence of set-open and uniform topologies. Proc. Steklov Inst. Math. Suppl. 3, 184–191 (2009)

    Article  MathSciNet  Google Scholar 

  19. Osipov, A.V.: The C-compact-open topology on function spaces. Topol. Appl. 159(13), 3059–3066 (2012)

    Article  MathSciNet  Google Scholar 

  20. Pichardo-Mendoza, R., Tamariz-Mascarúa, Á., Villegas-Rodríguez, H.: Pseudouniform topologies on C(X) given by ideals. Comment. Math. Univ. Carolin. 54(4), 557–577 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Sanchis, M.: A note on quasi-\(k\)-spaces. Rend. Istit. Mat. Univ. Trieste 30, 173–179 (1999)

    MathSciNet  MATH  Google Scholar 

  22. Shakhmatov, D.B.: A pseudocompact Tychonoff space all countable subsets of which are closed and \(C^*\)-embedded. Topol. Appl. 22, 139–144 (1986)

    Article  MathSciNet  Google Scholar 

  23. Shiraki, T.: A note on \(M\)-spaces. Proc. Jpn Acad. Ser. A Math. Sci. 43, 873–875 (1967)

    Article  MathSciNet  Google Scholar 

  24. Tkachuk, V.V., Wilson, R.G.: Cellular-compact spaces and their applications. Acta Math. Hungarica 159, 674–688 (2019)

    Article  MathSciNet  Google Scholar 

  25. Vaughan, J. E.: Countably compact and sequentially compact spaces, Handbook of Set-Theoretic Topology, (K. Kunen and J.E. Vaughan, eds.), North-Holland, Amsterdam, pp. 569–602 (1984)

Download references

Acknowledgements

We thank the referee for his careful and well-founded report by means of which we achieve that the current article has significant improvements over a previous version.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito exterior s/n, Ciudad Universitaria, CP 04510, Ciudad de México, México

    Juan Alberto Martínez-Cadena & Ángel Tamariz-Mascarúa

Authors
  1. Juan Alberto Martínez-Cadena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ángel Tamariz-Mascarúa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Alberto Martínez-Cadena.

Additional information

Communicated by Mohammad Koushesh.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Juan Alberto Martínez-Cadena acknowledges financial support from Programa de Becas Posdoctorales, UNAM-DGAPA, 2019–2020. The research of the second author was supported by grant CONACyT 220426.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martínez-Cadena, J.A., Tamariz-Mascarúa, Á. The Selectively Pseudocompact-Open Topology on C(X). Bull. Iran. Math. Soc. 47, 1611–1628 (2021). https://doi.org/10.1007/s41980-020-00462-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-020-00462-x

Keywords

Mathematics Subject Classification

Search

Navigation