Co-Hopf Space Structure on Closure Spaces

Year 2022, Volume: 14 Issue: 2, 321 - 330, 30.12.2022

Sibel Demiralp

https://doi.org/10.47000/tjmcs.1091039

Cited By: 2

https://izlik.org/JA24PU35LC

Abstract

By constructing Hopf costructures on closure spaces via homotopy, we give the concepts of closure Hopf cospace (CH-cospace) and closure Hopf cogroup (CH-cogroup). We then prove that retract and deformation retract of a CH-cospace are also a CH-cospace. We construct a Hopf costructure on a set with the help of the quotient closure operator. We also show that a closure space with the same homotopy type as a CH-cogroup is itself a CH-cogroup. We prove the existence of a covariant functor between the homotopy category of the pointed closure spaces ($\mathcal{CHC}$) and the category of groups and homomorphisms.

Keywords

Homotopy , closure space , Hopf cospace

References

• Adhikari, M.R., Rahaman, M., A study of some aspects of topological groups, Filomat, 21(1)(2007), 55–65.
• Arkowitz, M., Introduction to Homotopy Theory, Springer, New York, 2011.
• Boonpok, C., On continuous maps in closure spaces, General mathematics, 17(2)(2009), 127–134.
• Cech, E., Topological Spaces, Czechoslovak Acad. of Sciences, Prag, 1966.
• Ege, O., Karaca, I., Digital H-spaces, Proceeding of 3rd International Symposium on Computing in Science and Engineering, Kuadas-Turkey, October 24-25 (2013), 133–138.
• Ege, O., Karaca, I., Some properties of digital H-spaces, Turkish Journal of Electrical Engineering and Computer Sciences, 24(3)(2016), 1930–1941.
• Ege, O., Karaca, I., Digital co-Hopf spaces, Filomat, 34(8)(2020), 2705–2711.
• Eroglu, I., Guner, E., Separation axioms in Cech closure ordered spaces, Commun. Fac. Sci. Univ. Ank. Ser A1 Math. Stat, 65(2016,) 1–10.
• Lee, D.W., Digital H-spaces and actions in the pointed digital homotopy category, Applicable Algebra in Engineering, Communication and Computing, 31(2020), 149169.
• Mashhour, A.S., Ghanim, M.H., On closure spaces, Indian J. Pure Appl. Math, 14(6)(1983), 680–691.
• Park, K., On Sub-H-Groups of an H group and their duals, Journal of the Korean Mathematical Society, 6(1)(1969), 41–46.
• Rieser, A., Cech closure spaces:A unified framework for discrete and continuous homotopy, Topology and its Applications, 296(2021).