Acta Mathematica Vietnamica

, Volume 44, Issue 3, pp 711–722 | Cite as

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

  • Noorollah Jamali
  • Behrooz MashayekhyEmail author
  • Hamid Torabi
  • Seyyed Zeynal Pashaei
  • Mehdi Abdullahi Rashid


In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, \(\pi _{1}^{qtop}(X,x_{0})\) and \(\pi _{1}^{wh}(X,x_{0})\), respectively. In particular, we give necessary or sufficient conditions for equality and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the converse of some of these implications do not hold, in general.


Small loop transfer space Quasitopological fundamental group Whisker topology Topological group 

Mathematics Subject Classification (2010)

57M05 57M12 55P35 54H11 



The authors would like to thank the referee for the valuable comments and suggestions that helped to improve the manuscript.

Funding Information

This research was supported by a grant from Ferdowsi University of Mashhad-Graduate Studies (No. 38590).


  1. 1.
    Abdullahi Rashid, M., Mashayekhy, B., Torabi, H., Pashaei, S. Z.: On subgroups of topologized fundamental groups and generalized coverings. To appear in Bull. Iranian Math. Soc. arXiv:1602.07965
  2. 2.
    Bogley, W.A., Sieradski, A.J.: Universal path spaces. Preprint.
  3. 3.
    Brazas, J.: Semicoverings, coverings, overlays, and open subgroups of the quasitopological fundamental group. Topology Proc. 44, 285–313 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brazas, J.: The fundamental group as topological group. Topology Appl. 160(1), 170–188 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brazas, J., Fabel, P.: On fundamental groups with the quotient topology. J. Homotopy Relat. Struct. 10(1), 71–91 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brodskiy, N., Dydak, J., Labuz, B., Mitra, A.: Topological and uniform structures on universal covering spaces. arXiv:1206.0071
  7. 7.
    Calcut, J. S., McCarthy, J. D.: Discreteness and homogeneity of the topological fundamental group. Topology Proc. 34, 339–349 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Fabel, P.: Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology). Bull. Pol. Acad. Sci. Math. 59(1), 77–83 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fabel, P.: Compactly generated quasitopological homotopy groups with discontinuous multiplication. Topology Proc. 40, 303–309 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fischer, H., Zastrow, A.: Generalized universal covering spaces and the shape group. Fund. Math. 197, 167–196 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pakdaman, A., Torabi, H., Mashayekhy, B.: Small loop spaces and covering theory of non-homotopically Hausdorff spaces. Topology Appl. 158(6), 803–809 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pashaei, S. Z., Mashayekhy, B., Torabi, H., Abdullahi Rashid, M.: Small loop transfer spaces with respect to subgroups of fundamental groups. Topology Appl. 232, 242–255 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Spanier, E. H.: Algebraic Topology. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  14. 14.
    Torabi, H., Pakdaman, A., Mashayekhy, B.: On the Spanier groups and covering and semicovering spaces. arXiv:1207.4394
  15. 15.
    Torabi, H., Pakdaman, A., Mashayekhy, B.: Topological fundamental groups and small generated coverings. Math. Slovaca 65(5), 1153–1164 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Virk, Z.: Small loop spaces. Topology Appl. 157(2), 451–455 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Virk, Z., Zastrow, A.: The comparison of topologies related to various concepts of generalized covering spaces. Topology Appl. 170, 52–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran
  2. 2.Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic StructuresFerdowsi University of MashhadMashhadIran

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