Topological Group Construction in Proximity and Descriptive Proximity Spaces

Year 2024, , 206 - 216, 30.06.2024

Melih İs

https://doi.org/10.47000/tjmcs.1333562

Abstract

This paper introduces the topological group structure in proximity and especially descriptive proximity
spaces, that is, the concepts of proximal group and descriptive proximal group are introduced. In addition,
the concepts of homomorphism and isomorphism, which give important results in group theory, are discussed by interpreting the concepts of continuity in the theory of (descriptive) proximity

Keywords

Descriptive proximity, proximity, topological groups

References

• Cech, E., Topological Spaces, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1966.
• Di Concilio, A., Guadagni, C., Peters, J.F., Ramanna, S., Descriptive proximities. Properties and interplay between classical proximities and overlap, Math. Comput. Sci., 12(1)(2018), 91–106.
• Efremovic, V.A., The geometry of proximity I, Mat. Sb.(New Series), 31(73)(1952), 189–200.
• Hungerford, T., Algebra, Springer-Verlag, New York, 1974.
• Husain, T., Introduction to Topological Groups, W. B. Saunders Company, Philadelphia, 1966.
• İnan, E., Uçkun, M., Semitopological δ−groups, Hacet. J. Math. Stat., 52(1)(2023), 163–170.
• İs, M., Karaca, İ., Some properties on proximal homotopy theory, Filomat, 38(9)(2024), 3137–3156.
• Kuratowski, C., Topologie. I, Panstwowe Wydawnictwo Naukowe, Warsaw, 1958.
• Leader, S., On products of proximity spaces, Math. Ann., 154(1964), 185–194.
• Lodato, M.W., On topologically induced generalized proximity relations, Proc. Amer. Math. Soc., 15(1964), 417–422.
• Maheswari, A., Ilango, G., Topological near groups, Asian Res. J. Math., 18(12)(2022), 46–56.
• Maheswari, A., Ilango, G., Topological near homomorphisms, J. Res. Appl. Math., 9(2)(2023), 46–54.
• Naimpally, S.A., Warrack, B.D., Proximity Spaces, Cambridge Tract in Mathematics No. 59, Cambridge University Press, Cambridge, UK, Paperback 2008, MR0278261 1970.
• Naimpally, S.A., Peters, J.F., Topology With Applications. Topological Spaces via Near and Far, World Scientific, Singapore, 2013.
• Peters, J.F., Near sets. General theory about nearness of objects, Appl. Math. Sci., 1(53)(2007), 2609–2629.
• Peters, J.F., Near sets. Special theory about nearness of objects, Fundam. Inform., 75(1-4)(2007), 407–433.
• Peters, J.F., Near sets: An introduction, Math. Comput. Sci., 7(1)(2013), 3–9.
• Peters, J.F., Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces (Vol. 63), Springer Science & Business Media, 2014.
• Peters, J.F., Foundations of Computer Vision: Computational Geometry, Visual Image Structures and Object Shape Detection (Vol. 124), Springer Science & Business Media, 2017.
• Peters J.F., Vergili, T., Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering ond Jordan Curve Theorems, Appl. Gen. Topol., 24(1)(2023), 25–45.
• Peters, J.F. Vergili, T., Proximity Space Categories. Results For Proximal Lyusternik-Schnirel’man, Csaszar And Bornology Categories., Submitted, 2023.
• Riesz, F., Stetigkeitsbegriff und abstrakte mengenlehre, Atti del IV Congresso Internazionale dei Matematici II, 1908.
• Smirnov, Y.M., On proximity spaces, Mat. Sb.(New Series), 31(73)(1952), 543–574. English Translation: Amer. Math. Soc. Trans.: Series 2, 38(1964), 5–35.
• Wallace, A.D., Separation space, Ann. Math., (1941), 687–697.