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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
Article
  • 24 Downloads

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.

Keywords

Small loop transfer space Quasitopological fundamental group Whisker topology Topological group 

Mathematics Subject Classification (2010)

57M05 57M12 55P35 54H11 

Notes

Acknowledgements

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).

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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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