Research Article
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Topological Group Construction in Proximity and Descriptive Proximity Spaces

Year 2024, Volume: 16 Issue: 1, 206 - 216, 30.06.2024
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

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.
Year 2024, Volume: 16 Issue: 1, 206 - 216, 30.06.2024
https://doi.org/10.47000/tjmcs.1333562

Abstract

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.
There are 24 citations in total.

Details

Primary Language English
Subjects Topology
Journal Section Articles
Authors

Melih İs 0000-0002-1272-6701

Publication Date June 30, 2024
Published in Issue Year 2024 Volume: 16 Issue: 1

Cite

APA İs, M. (2024). Topological Group Construction in Proximity and Descriptive Proximity Spaces. Turkish Journal of Mathematics and Computer Science, 16(1), 206-216. https://doi.org/10.47000/tjmcs.1333562
AMA İs M. Topological Group Construction in Proximity and Descriptive Proximity Spaces. TJMCS. June 2024;16(1):206-216. doi:10.47000/tjmcs.1333562
Chicago İs, Melih. “Topological Group Construction in Proximity and Descriptive Proximity Spaces”. Turkish Journal of Mathematics and Computer Science 16, no. 1 (June 2024): 206-16. https://doi.org/10.47000/tjmcs.1333562.
EndNote İs M (June 1, 2024) Topological Group Construction in Proximity and Descriptive Proximity Spaces. Turkish Journal of Mathematics and Computer Science 16 1 206–216.
IEEE M. İs, “Topological Group Construction in Proximity and Descriptive Proximity Spaces”, TJMCS, vol. 16, no. 1, pp. 206–216, 2024, doi: 10.47000/tjmcs.1333562.
ISNAD İs, Melih. “Topological Group Construction in Proximity and Descriptive Proximity Spaces”. Turkish Journal of Mathematics and Computer Science 16/1 (June 2024), 206-216. https://doi.org/10.47000/tjmcs.1333562.
JAMA İs M. Topological Group Construction in Proximity and Descriptive Proximity Spaces. TJMCS. 2024;16:206–216.
MLA İs, Melih. “Topological Group Construction in Proximity and Descriptive Proximity Spaces”. Turkish Journal of Mathematics and Computer Science, vol. 16, no. 1, 2024, pp. 206-1, doi:10.47000/tjmcs.1333562.
Vancouver İs M. Topological Group Construction in Proximity and Descriptive Proximity Spaces. TJMCS. 2024;16(1):206-1.