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On Spaces Star Determined by Compact Metrizable Subspaces

  • Wei-Feng XuanEmail author
  • Yan-Kui Song
Original Paper
  • 7 Downloads

Abstract

A space X is said to be star determined by compact metrizable subspaces (star-CM for short) if for any open cover \({\mathcal {U}}\) of X there is a compact and metrizable subspace \(Y \subset X\) such that \({\text {St}}(Y, {\mathcal {U}})=X\). This notation of star-CM was introduced by van Mill, Tkachuk and Wilson in (Topol Appl 154:2127–2134, 2007). In this paper, we investigate the relations between star-CM spaces and related spaces, and study topological properties of star-CM spaces. We also establish a cardinal theorem for star-CM spaces with symmetric g-functions.

Keywords

Star-CM Star countable Star compact g-Function 

Mathematics Subject Classification

54D20 54E35 

Notes

Acknowledgements

We would like to thank the referee for his (or her) valuable remarks and suggestions which greatly improved the paper.

References

  1. 1.
    Arhangel’skii, A.V., Burke, D.K.: Spaces with a regular \(G_\delta \)-diagonal. Topol. Appl. 153(11), 1917–1929 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Engelking, R.: General Topology. Revised and completed edition. Heldermann Verlag, Berlin (1989)Google Scholar
  3. 3.
    Good, C., Jennings, D., Mohamad, A.M.: Symmetric \(g\)-functions. Topol. Appl. 134, 111–122 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Hodel, R.: Cardinal functions I. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 1–61. North-Holland, Amsterdam (1984)Google Scholar
  5. 5.
    Ikenaga, S.: Topological concept between Lindelöf and Pseudo-Lindelöf. Res. Rep. Nara Natl. Coll. Technol. 26, 103–108 (1990)Google Scholar
  6. 6.
    Matveev, M.V.: A survey on star covering properties. Topology Atlas (preprint) (1998)Google Scholar
  7. 7.
    Porter, J.R., Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York (1980)zbMATHGoogle Scholar
  8. 8.
    Song, Y.K.: On \({\cal{K}}\)-starcompact spaces. Bull. Malays. Math. Sci. Soc. 30(1), 59–64 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Sun, S.-H.: A natural generalization of Chaber’s theorem (English Ed.). Kexue Tongbao 30(11), 1431–1436 (1985)MathSciNetzbMATHGoogle Scholar
  10. 10.
    van Douwen, E.K., Reed, G.M., Roscoe, A.W., Tree, I.J.: Star covering properties. Topol. Appl. 39(1), 71–103 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    van Mill, J., Tkachuk, V.V., Wilson, R.G.: Classes defined by stars and neighbourhood assignments. Topol. Appl. 154, 2127–2134 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Xuan, W.F., Shi, W.X.: Some results on star-covering property and cardinal inequality. Q & A in Gen. Top. 34, 43–48 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.School of statistics and MathematicsNanjing Audit UniversityNanjingChina
  2. 2.Institute of Mathematics, School of Mathematical ScienceNanjing Normal UniversityNanjingChina

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