Acta Mathematica Hungarica

, Volume 159, Issue 2, pp 674–688 | Cite as

Cellular-compact spaces and their applications

  • V. V. TkachukEmail author
  • R. G. Wilson


We introduce the notion of a cellular-compact space and prove that cellular compactness is a nice property that implies cellular Lindelöfness. The class of cellular-compact spaces is preserved by continuous images and finite unions, as well as by regular closed subsets and extensions. It is established that cellular-compact spaces must be pseudocompact but not necessarily countably compact. We prove that first countable cellular-compact regular spaces are countably compact and their cardinality does not exceed \({2^{\omega}}\). We also show that a collectionwise normal Fréchet–Urysohn cellular-compact space need not be compact and there exist Fréchet–Urysohn cellular-compact spaces that do not have a dense countably compact subspace.

Key words and phrases

cellular-compact space cellular-Lindelöf space weakly Lindelöf space disjoint local \(\pi\)-base first countable space maximal cellular-compact space 

Mathematics Subject Classification

primary 54D20 54D99 secondary 54D55 


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  1. 1.
    O. T. Alas, L. R. Junqueira, and R. G. Wilson, On linearly H-closed spaces, Topology Appl., 258 (2019), 161–171.MathSciNetCrossRefGoogle Scholar
  2. 2.
    O. T. Alas, V. V. Tkachuk, and R. G. Wilson, Closures of discrete sets often reflect global properties, Topology Proc., 25 (2000), 27–44.MathSciNetzbMATHGoogle Scholar
  3. 3.
    A. V. Arhangel'skii, Structure and classification of topological spaces and cardinal invariants, Uspehi Mat. Nauk, 33 (1978), 29–84 (in Russian).MathSciNetGoogle Scholar
  4. 4.
    M. Baillif, Notes on linearly H-closed spaces and od-selection principles, Topology Proc., 54 (2019), 109–124.MathSciNetzbMATHGoogle Scholar
  5. 5.
    B. Balcar and P. Vojtáš, Almost disjoint refinement of families of subsets of N, Proc. Amer. Math. Soc., 79 (1980), 465–470.MathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Bell, J. Ginsburg, and G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math., 79 (1978), 37–45.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Bella, Observations on some cardinality bounds, Topology Appl., 228 (2017), 355–362.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Bella and S. Spadaro, On the cardinality of almost discretely Lindelöf spaces, Monatsh. Math., 186 (2018), 345–353.MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Bella and S. Spadaro, Cardinal invariants of cellular-Lindelöf spaces, Revista de la Real Acad. Ciencias Exactas, Ser. A. Mat. (RACSAM) (in print).Google Scholar
  10. 10.
    E. K. van Douwen, Existence and applications of remote points, Bull. Amer. Math. Soc., 84 (1978), 161–163.MathSciNetCrossRefGoogle Scholar
  11. 11.
    A. Dow, M. G. Tkachenko, V. V. Tkachuk, and R. G. Wilson, Topologies generated by discrete subspaces, Glasnik Mat., 37 (2002), 187–210.MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. Engelking, General Topology, PWN (Warszawa, 1977).Google Scholar
  13. 13.
    R.E. Hodel, Cardinal Functions. I, in: Handbook of Set-Theoretic Topology, K. Kunen, J. E. Vaughan, eds., North Holland (Amsterdam, 1984), pp. 1–61.Google Scholar
  14. 14.
    J. R. Porter, R. M. Stephenson, Jr., and R. G. Woods, Maximal feebly compact spaces, Topology Appl., 52 (1993), 203–219.MathSciNetCrossRefGoogle Scholar
  15. 15.
    V. V. Tkachuk, Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 50 (1988), 139–156.MathSciNetzbMATHGoogle Scholar
  16. 16.
    V. V. Tkachuk, A C p-theory Problem Book. Topological and Function Spaces, Springer (New York, 2011).CrossRefGoogle Scholar
  17. 17.
    V. V. Tkachuk, A C p-theory Problem Book. Special Features of Function Spaces, Springer (New York, 2014).Google Scholar
  18. 18.
    V. V. Tkachuk, A C p-Theory Problem Book. Compactness in Function Spaces, Springer (New York, 2015).CrossRefGoogle Scholar
  19. 19.
    V. V. Tkachuk, Weakly linearly Lindelöf spaces revisited, Topology Appl., 256 (2019), 128–135.MathSciNetCrossRefGoogle Scholar
  20. 20.
    W.-F. Xuan and Y.-K. Song, On cellular-Lindelöf spaces, Bull. Iranian Math. Soc., 44 (2018), 1485–1491.MathSciNetCrossRefGoogle Scholar
  21. 21.
    W.-F. Xuan and Y.-K. Song, A study of cellular-Lindelöf spaces, Topology Appl., 251 (2019), 1–9.MathSciNetCrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma MetropolitanaMexico D.F.Mexico

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