Sequential Rectifiable Spaces of Countable \(\mathrm {cs}^*\)-Character

  • Taras Banakh
  • Dušan Repovš


We prove that each non-metrizable sequential rectifiable space X of countable \(\mathrm {cs}^*\)-character contains a clopen rectifiable submetrizable \(k_\omega \)-subspace H and admits a disjoint cover by open subsets homeomorphic to clopen subspaces of H. This implies that each sequential rectifiable space of countable \(\mathrm {cs}^*\)-character is either metrizable or a topological sum of submetrizable \(k_\omega \)-spaces. Consequently, X is submetrizable and paracompact. This answers a question of Lin and Shen posed in 2011.


Rectifiable space Sequential space \(k_\omega \)-Space \(\mathrm {cs}^*\)-Character Topological loop Topological left-loop Topological lop 

Mathematics Subject Classification

54D55 54D50 54H10 22A22 22A30 


  1. 1.
    Arhangel’skii, A.: Topological invariants in algebraic environment, In: Recent Progress in General Topology, II, 1–57, North-Holland, Amsterdam, (2002)Google Scholar
  2. 2.
    Banakh, T.: On topological groups containing a Fréchet-Urysohn fan. Matem. Studii 9(2), 149–154 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banakh, T., Zdomskyy, L.: The topological structure of (homogeneous) spaces and groups with countable \({\rm cs}^*\)-character. Appl. Gen. Topol. 5(1), 25–48 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burke, D.: Covering Properties, Handbook of Set-Theoretic Topology, pp. 347–422. North-Holland, Amsterdam (1984)CrossRefGoogle Scholar
  5. 5.
    van Douwen, E.: The integers and topology. In: Kunen, K., Vaughan, J. (eds.) Handbook of Set-Theoretic Topology, pp. 111–167. North-Holland, Amsterdam (1984)CrossRefGoogle Scholar
  6. 6.
    Engelking, R.: General Topology. Heldermann Verlag, Berlin (1989)zbMATHGoogle Scholar
  7. 7.
    Gruenhage, G.: Generalized Metric Spaces, Handbook of Set-Theoretic Topology, pp. 423–501. North-Holland, Amsterdam (1984)CrossRefGoogle Scholar
  8. 8.
    Gruenhage, G., Tsaban, B., Zdomskyy, L.: Sequential properties of function spaces with the compact-open topology. Topol. Appl. 158(3), 387–391 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gruenhage, G.: Generalized metric spaces and metrization, Recent Progress in General Topology (Prague, 1991), 239–274. North-Holland, Amsterdam (1992)Google Scholar
  10. 10.
    Gul’ko, A.: Rectifiable spaces. Topol. Appl. 68(2), 107–112 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hofmann, K., Martin, J.: Topological left-loops. Topol. Proc. 39, 185–194 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hofmann, K.H., Strambach, K.: Topological and analytic loops. In: Chein, O., Pflugfelder, H.O., Smith J.D.H. (eds.) Quasigroups and Loops: Theory and Applications, pp. 205–262. Heldermann Verlag, Berlin (1990)Google Scholar
  13. 13.
    Lin, F.: Metrizability of rectifiable spaces. Bull. Malays. Math. Sci. Soc. (2) 36(4), 1099–1107 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Lin, F., Liu, C., Lin, S.: A note on rectifiable spaces. Topol. Appl. 159(8), 2090–2101 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lin, F., Shen, R.: On rectifiable spaces and paratopological groups. Topol. Appl. 158(4), 597–610 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin, S.: A note on the Arens’ space and sequential fan. Topol. Appl. 81, 185–196 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Robinson, D.: A Course in the Theory of Groups. Springer, Berlin (1996)CrossRefGoogle Scholar
  18. 18.
    Uspenskii, V.: Topological groups and Dugundji compact spaces. Mat. Sb. 180(8), 1092–1118 (1989)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2016

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsIvan Franko National University of LvivLvivUkraine
  2. 2.Institute of Mathematics, Jan Kochanowski University in KielceKielcePoland
  3. 3.Faculty of Education, and Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia

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