Acta Mathematica Hungarica

, Volume 159, Issue 1, pp 229–245 | Cite as

Cardinal inequalities for S(n)-spaces

  • I. S. GotchevEmail author


Hajnal and Juhász [9] proved that if X is a T1-space, then \({|X| \leq 2^{s(X)\psi(X)}}\), and if X is a Hausdorff space, then \({|X| \leq 2^{c(X)\chi(X)}}\) and \({|X| \leq 2^{2^{s(X)}}}\). Schröder sharpened the first two estimations by showing that if X is a Hausdorff space, then \({|X| \leq 2^{Us(X)\psi_c(X)}}\), and if X is a Urysohn space, then \({|X| \leq 2^{Uc(X)\chi(X)}}\).

In this paper, for any positive integer n and some topological spaces X, we define the cardinal functions \({\chi_n(X), \psi_n(X), s_n(X)}\), and cn(X) called respectively S(n)-character, S(n)-pseudocharacter, S(n)-spread, and S(n)-cellularity and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of S(n)-spaces. We recall that the S(1)-spaces are exactly the Hausdorff spaces and the S(2)-spaces are exactly the Urysohn spaces.

Key words and phrases

cardinal function S(n)-space S(n)-character S(n)-pseudocharacter S(n)-discrete S(n)-spread S(n)-cellularity 

Mathematics Subject Classification

primary 54A25 54D10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author is grateful to the anonymous referee for very careful reading of the paper and for several valuable suggestions.


  1. 1.
    O. T. Alas and Lj. D. R. Kočinac, More cardinal inequalities on Urysohn spaces, Math. Balkanica (N.S.), 14 (2000), 247–251Google Scholar
  2. 2.
    Charlesworth, A.: On the cardinality of a topological space. Proc. Amer. Math. Soc. 66, 138–142 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dikranjan, D., Giuli, E.: \(S(n)\)-\(\theta \)-closed spaces. Topology Appl. 28, 59–74 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and its Applications, 346, Kluwer Academic Publishers Group (Dordrecht, 1995)Google Scholar
  5. 5.
    Dikranjan, D., Watson, S.: The category of \(S(\alpha )\)-spaces is not cowellpowered. Topology Appl. 61, 137–150 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, revised ed., Heldermann Verlag (Berlin, 1989)Google Scholar
  7. 7.
    Gotchev, I.S.: Cardinal inequalities for Urysohn spaces involving variations of the almost Lindelöf degree. Serdica Math. J. 44, 195–212 (2018)MathSciNetGoogle Scholar
  8. 8.
    I. S. Gotchev and Lj. D. R. Kočinac, More on the cardinality of \(S(n)\)-spaces, Serdica Math. J., 44 (2018), 227 – 242Google Scholar
  9. 9.
    A. Hajnal and I. Juhász, Discrete subspaces of topological spaces, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math., 29 (1967), 343–356Google Scholar
  10. 10.
    R. E. Hodel, Cardinal functions. I, in: Handbook of Set-Theoretic Topology, (K. Kunen and J. E. Vaughan, eds.) North-Holland (Amsterdam, 1984), pp. 1–61Google Scholar
  11. 11.
    I. Juhász, Cardinal Functions in Topology—Ten Years Later, Mathematical Centre Tracts, No. 123, Mathematisch Centrum (Amsterdam, 1980)Google Scholar
  12. 12.
    Pol, R.: Short proofs of two theorems on cardinality of topological spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22, 1245–1249 (1974)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Porter, J.R., Votaw, C.: \(S(\alpha )\) spaces and regular Hausdorff extensions. Pacific J. Math. 45, 327–345 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schröder, J.: Urysohn cellularity and Urysohn spread. Math. Japon. 38, 1129–1133 (1993)MathSciNetzbMATHGoogle Scholar
  15. 15.
    N. V. Veličko, \(H\)-closed topological spaces, Mat. Sb. (N.S.), 70 (1966), 98–112Google Scholar
  16. 16.
    Viglino, G.: \(\bar{T}_{n}\)-spaces. Kyungpook Math. J. 11, 33–35 (1971)MathSciNetzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCentral Connecticut State UniversityNew BritainUSA

Personalised recommendations