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Notes on selectively 2-star-ccc spaces

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Bal and Kočinac (Topol Appl, https://doi.org/10.1016/j.topol.2020.107184, 2020) introduced the class of selectively k-star-ccc spaces. A space X is called selectivelyk-star-ccc if for every open cover \(\mathcal U\) of X and every sequence \((\mathcal A_n: n\in \omega )\) of maximal pairwise disjoint open families in X there exists a sequence \((A_n: n\in \omega )\) such that \(A_n\in \mathcal A_n\) for every \(n\in \omega \) and \({\text {St}}^{{\text {k}}}(\bigcup _{n\in \omega }A_n, \mathcal U)=X\), where \(k\in \omega {\setminus }\{0\}\). In this paper, we give several results on selectively 2-star-ccc spaces and state a number of open questions for further investigation.

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References

  1. Alas, O.T., Wilson, R.G.: Properties related to star countability and star finiteness. Topol. Appl. 221, 432–439 (2017)

    Article  MathSciNet  Google Scholar 

  2. Alas, O.T., Junqueira, L., van Mill, J., Tkachuk, V.V., Wilson, R.G.: On the extent of star countable spaces. Cent. Eur. J. Math. 9(3), 603–615 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bal, P., Kočinac, L.D.R.: On selectively star-ccc spaces. Topol. Appl. (2020). https://doi.org/10.1016/j.topol.2020.107184. (, Art. No. 107184, in press )

  4. Bella, A., Spadaro, S.: On the cardinality of almost discretely Lindelöf spaces. Monatsh Math. 186(2), 345–353 (2018)

    Article  MathSciNet  Google Scholar 

  5. Bell, M.G.: First countable pseudocompactifications. Topol. Appl. 21(2), 159–166 (1985)

    Article  MathSciNet  Google Scholar 

  6. Engelking, R.: General Topology, Revised and completed edition edn. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  7. Hodel, R.: Cardinal functions I. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 1–61. North-Holland, Amsterdam (1984)

    Google Scholar 

  8. Junnila, H.J.K.: On countability of point-finite families of sets. Can. J. Math. 31, 673–679 (1979)

    Article  MathSciNet  Google Scholar 

  9. Kočinac, L.D.R.: Star-Menger and related spaces. Publ. Math. Debrecen. 55(3–4), 421–431 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Kočinac, L.D.R.: Star-Menger and related spaces II. Filomat 13, 129–140 (1999)

    MathSciNet  MATH  Google Scholar 

  11. Kočinac, L.D.R.: Star selection principles: a survey. Khayyam J. Math. 1, 82–106 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Kočinac, L.D.R.: Selected results on selection principles. In: Proceedings of the Third Seminar on Geometry and Topology (July 15–17, 2004, Tabriz, Iran), pp. 71–104

  13. Kočinac, L.D.R., Scheepers, M.: Combinatorics of open covers (VII): groupability. Fund. Math. 179, 131–155 (2003)

    Article  MathSciNet  Google Scholar 

  14. Kuratowski, K.: Topology, vol. I. Academic Press, New York (1966)

    MATH  Google Scholar 

  15. Lutzer, D.J.: Pixley–Roy topology. Topol. Proc. 1, 139–158 (1978)

    MathSciNet  MATH  Google Scholar 

  16. Menger, K.: Einige Überdeckungssätze der Punktmengenlehre. Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien), 133, 421–444 (1924)

  17. Przymusiński, T., Tall, F.D.: The undecidability of the existence of a non-separable normal Moore space satisfying the countable chain condition. Fund. Math. 85, 291–297 (1974)

    Article  MathSciNet  Google Scholar 

  18. Rothberger, F.: Eine Vershärfung der Eigenschaft C. Fund. Math. 30, 50–55 (1938)

    Article  Google Scholar 

  19. Rojas-Sánchez, A.D., Tamariz-Mascarúa, Á.: Spaces with star countable extent. Comment. Math. Univ. Carolin. 57(3), 381–395 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Sakai, M.: Cardinal functions of Pixley–Roy hyperspaces. Topol. Appl. 159(13), 3080–3088 (2012)

    Article  MathSciNet  Google Scholar 

  21. Scheepers, M.: The combinatorics of open covers (I): Ramsey theory. Topol. Appl. 69, 31–62 (1996)

    Article  MathSciNet  Google Scholar 

  22. Song, Y.K., Xuan, W.F.: A note on selectively star-ccc spaces. Topol. Appl. 263, 343–349 (2019)

    Article  MathSciNet  Google Scholar 

  23. Song, Y.K., Xuan, W.F.: More on selectively star-ccc spaces. Topol. Appl. 268, 106905 (2019)

    Article  MathSciNet  Google Scholar 

  24. Tall, F.D.: Normality versus collectionwise normality. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 685–732. North-Holland, Amsterdam (1984)

    Chapter  Google Scholar 

  25. van Douwen, E.K.: The Pixley–Roy topology on spaces of subsets. Set Theoretic Topology, pp. 111–134. Academic Press, New York (1977)

    Chapter  Google Scholar 

  26. van Mill, J., Tkachuk, V.V., Wilson, R.G.: Classes defined by stars and neighbourhood assignments. Topol. Appl. 154, 2127–2134 (2007)

    Article  MathSciNet  Google Scholar 

  27. Xuan, W.F., Song, Y.K.: On properties related to star countability. Topol. Appl. 247, 20–28 (2018)

    Article  MathSciNet  Google Scholar 

  28. Xuan, W.F., Song, Y.K.: A study of selectively star-ccc spaces. Topol. Appl. 273, 107103 (2020)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, China

    Wei-Feng Xuan

  2. Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China

    Yan-Kui Song

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  1. Wei-Feng Xuan
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  2. Yan-Kui Song
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Correspondence to Wei-Feng Xuan.

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Wei-Feng Xuan is supported by NSFC project 11801271. Yan-Kui Song is supported by NSFC project 11771029.

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Xuan, WF., Song, YK. Notes on selectively 2-star-ccc spaces. RACSAM 114, 155 (2020). https://doi.org/10.1007/s13398-020-00884-6

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