Co-Hopf Space Structure on Closure Spaces

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,

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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).