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Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 109–123 | Cite as

On cardinality bounds for \(\theta^n\)-Urysohn spaces

  • F. A. BasileEmail author
  • N. Carlson
  • J. Porter
Article
  • 13 Downloads

Abstract

We introduce the class of \(\theta^n\)-Urysohn spaces and the \(n\)-\(\theta\)-closure operator. \(\theta^n\)-Urysohn spaces generalize the notion of a Urysohn space and we consider their relationship with S(n)-spaces, studied in [9], [14] and [18]. We estabilish bounds on the cardinality of these spaces and cardinality bounds if the space is additionally homogeneous.

Key words and phrases

Urysohn homogeneous space \(\theta\)-closure \(\theta^n\)-closure Lindelöf degree 

Mathematics Subject Classification

54A25 54D10 

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Notes

Acknowledgements

The authors are very grateful to the anonymous reviewer for valuable comments and suggestions to improve the quality of the paper and for suggesting Question 1. The first author would like to thank also Professor I. Gotchev for good conversations relating \(\theta^n\)-Urysohn spaces and S(n)-spaces.

References

  1. 1.
    Basile, F.A., Bonanzinga, M., Carlson, N.: Variations on known and recent cardinality bounds. Topology Appl. 240, 228–237 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Basile, F.A., Carlson, N.: On the cardinality of Urysohn spaces and weakly H-closed spaces. Math. Bohemica 144, 325–336 (2019)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bella, A., Cammaroto, F.: On the cardinality of Urysohn spaces. Canad. Math. Bull. 31, 153–158 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonanzinga, M., Carlson, N., Cuzzupé, M.V., Stavrova, D.: More on the cardinality of a topological space. App. Gen. Top. 19, 269–280 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cammaroto, F., Catalioto, A., Porter, J.: On the cardinality of Urysohn spaces. Topology Appl. 160, 1862–1869 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cammaroto, F., Catalioto, A., Porter, J.: Cardinal functions \(F_{\theta }(X)\) and \(t_{\theta }(X)\) for \(H\)-closed spaces. Quest. Math. 37, 309–320 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    F. Cammaroto and Lj. Kočinac, On \(\theta \)-tightness, Facta Universitatis (Niš), Ser. Math. Inform., 8 (1993), 77–85Google Scholar
  8. 8.
    Carlson, N., Ridderbos, G.J.: Partition relations and power homogeneity. Topology Proc. 32, 115–124 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dikranjan, D., Giuli, E.: \(S(n)\text{-}\theta \)-closed spaces. Topology Appl. 28, 59–74 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    R. Engelking, General Topology, Heldermann Verlag (1989)Google Scholar
  11. 11.
    Gotchev, I.S.: Cardinal inequalities for Urysohn spaces involving variations of the almost Lindelöf degree. Serdica Math. J. 44, 195–212 (2018)MathSciNetGoogle Scholar
  12. 12.
    Hodel, R.E.: Arhangel'skii's solution to Alexandroff's problem: A survey. Topology Appl. 153, 2199–2217 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Osipov, On the cardinality of \(S(n)\)-spaces, arXiv:1809.09587
  14. 14.
    Porter, J.R., Votaw, C.: \(S(\alpha )\)-spaces and regular Hausdorff extensions. Pacific J. Math. 45, 327–345 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Schröder, J.: Urysohn cellularity and Urysohn spread. Math. Japonica 38, 1129–1133 (1993)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Shu-Hao, S.: Two new topological cardinal inequality. Proc. Amer. Math. Soc. 104, 313–316 (1988)MathSciNetCrossRefGoogle Scholar
  17. 17.
    N. V. Veličko, H-closed topological spaces, Mat. Sb. (N.S.), 70 (112) (1966), 98–112 (in Russian)Google Scholar
  18. 18.
    Viglino, G.: \(\overline{T}_{n}\)-spaces. Kyungpook Math. J. 11, 33–35 (1971)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Willard, S., Dissanayake, U.N.B.: The almost Lindelöf degree. Canad. Math. Bull. 27, 452–455 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Physics and Earth SciencesUniversity of MessinaMessinaItaly
  2. 2.Department of MathematicsCalifornia Lutheran University, Thousand OaksCaliforniaUSA
  3. 3.Department of MathematicsUniversity of KansasLawrenceUSA

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