On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

Abstract

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.

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References

  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. http://oregonstate.edu/bogleyw/

  3. 3.

    Brazas, J.: Semicoverings, coverings, overlays, and open subgroups of the quasitopological fundamental group. Topology Proc. 44, 285–313 (2014)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Brazas, J.: The fundamental group as topological group. Topology Appl. 160(1), 170–188 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Brazas, J., Fabel, P.: On fundamental groups with the quotient topology. J. Homotopy Relat. Struct. 10(1), 71–91 (2015)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Fabel, P.: Compactly generated quasitopological homotopy groups with discontinuous multiplication. Topology Proc. 40, 303–309 (2012)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Fischer, H., Zastrow, A.: Generalized universal covering spaces and the shape group. Fund. Math. 197, 167–196 (2007)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Spanier, E. H.: Algebraic Topology. McGraw-Hill, New York (1966)

    Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Virk, Z.: Small loop spaces. Topology Appl. 157(2), 451–455 (2010)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

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

Funding

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

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Correspondence to Behrooz Mashayekhy.

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Jamali, N., Mashayekhy, B., Torabi, H. et al. On Topologized Fundamental Groups with Small Loop Transfer Viewpoints. Acta Math Vietnam 44, 711–722 (2019). https://doi.org/10.1007/s40306-018-0264-1

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Keywords

  • Small loop transfer space
  • Quasitopological fundamental group
  • Whisker topology
  • Topological group

Mathematics Subject Classification (2010)

  • 57M05
  • 57M12
  • 55P35
  • 54H11