SavedAdd your journal

Journals / Fundamentals of Contemporary Mathematical Sciences / 2026 / Volume: 7 Issue: 2

Proximal Lusternik-Schnirelmann, Császár and Bornology Categories

James F. Peters(University of Manitoba)0000-0002-1026-4638
Tane Vergili(KARADENİZ TEKNİK ÜNİVERSİTESİ)0000-0003-1821-6697
Ebubekir İnan(KAYSERI UNIVERSITY)0000-0002-7269-8941
Pages
—
Publication Date
2026-07-31
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Several other results are also given, namely, Prox has all nonempty products and coproducts and ProxBor has all nonempty coproducts.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This paper introduces new forms of proximity space categories, namely, proxLS and dproxLS, spatial and descriptive proximity refinements of the Lusternik-Schnirelmann category LS and desProx, which is a descriptive proximity extension of the Császár Prox category.
📊
Key Findings & Data:A main result given here is that a proximity space (X, δ) is proximally contractible if and only if the proximal Lusternik-Schnirelmann category LSδ(X) = 1.
💡
Conclusion & Impact:Several other results are also given, namely, Prox has all nonempty products and coproducts and ProxBor has all nonempty coproducts.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Cech proximity · descriptive proximity · Lyusternik-Scnirelmann category

Abstract

This paper introduces new forms of proximity space categories, namely, proxLS and dproxLS, spatial and descriptive proximity refinements of the Lusternik-Schnirelmann category LS and desProx, which is a descriptive proximity extension of the Császár Prox category. A main result given here is that a proximity space (X, δ) is proximally contractible if and only if the proximal Lusternik-Schnirelmann category LSδ(X) = 1. Several other results are also given, namely, Prox has all nonempty products and coproducts and ProxBor has all nonempty coproducts.

🔖

Cite This Article

Peters, J. F., Vergili, T., İnan, E. (2026). Proximal Lusternik-Schnirelmann, Császár and Bornology Categories. Fundamentals of Contemporary Mathematical Sciences, Volume: 7 Issue: 2. https://doi.org/10.54974/fcmathsci.1703891

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (29)

  1. Borat A., Vergili T., Digital Lusternik-Schnirelmann category, Turkish Journal of Mathematics, 42(4), 1845-1852, 2018.
  2. Borsuk K., On the Lusternik-Schnirelmann category in the theory of shape, Fundamenta Mathematicae, 99(1), 35-42, 1978.
  3. Bunke U., Engel A., Homotopy Theory with Bornological Coarse Spaces, Springer, 2020.
  4. Čech E., Topological Spaces, John Wiley & Sons Ltd., 1966.
  5. Cornea O., Lupton G., Oprea J., Tanré D., Lusternik-Schnirelmann Category, Mathematical Surveys and Monographs, 103, American Mathematical Society, Providence, RI, 2003.
  6. Császár Á., Proximities, screens, merotopies, uniformities. I, Acta Mathematica Hungarica, 49(3-4), 459-479, 1987.
  7. Di Concilio A., Guadagni C., Peters J.F., Ramanna S., Descriptive proximities. Properties and interplay between classical proximities and overlap, Mathematics in Computer Science, 12(1), 91-106, 2018.
  8. Efremovič V.A., The geometry of proximity I, Matematicheskii Sbornik, 31(73)(1), 189-200, 1952.
  9. Félix Y., Halperin S., Thomas J.-C., Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2001.
  10. Fox R.H., On the Lusternik-Schnirelmann category, Annals of Mathematics, 42(2), 333-370, 1941.
  11. Grzegrzolka P., Coproducts of proximity spaces, Afrika Matematika, 31, 751-770, 2020.
  12. Hausdorff F., Grundzüge der Mengenlehre, Veit and Company, 1914.
  13. Hilbert D., Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, 1912; reprinted by Chelsea Publishing Company, 1953.
  14. Hilton P.J., The Hopf invariant and homotopy groups of spheres, Proceedings of the Cambridge Philosophical Society, 48, 547-554, 1952.
  15. Lusternik L.A., Schnirelmann L.G., Méthodes Topologiques dans les Problèmes Variationnels. I. Espaces à un Nombre Fini de Dimensions, Hermann, 1934.
  16. Lusternik L.A., Schnirelmann L.G., Topological methods in variational problems and their application to the differential geometry of surfaces, Uspekhi Matematicheskikh Nauk, 2(1)(17), 166-217, 1947.
  17. Naimpally S.A., Warrack B.D., Proximity Spaces, Cambridge Tracts in Mathematics and Mathematical Physics, 59, Cambridge University Press, 1970.
  18. Naimpally S.A., Peters J.F., Topology with Applications: Topological Spaces via Near and Far, World Scientific, 2013.
  19. Peters J.F., Near sets. Special theory about nearness of objects, Fundamenta Informaticae, 75, 407-433, 2007.
  20. Peters J.F., Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Intelligent Systems Reference Library, 63, Springer, 2014.
  21. Peters J.F., Vergili T., Descriptive fixed set properties for ribbon complexes, arXiv:2007.04394, 2020.
  22. Peters J.F., Vergili T., Fixed point property of amenable planar vortexes, Applied General Topology, 22(2), 385-397, 2021.
  23. Peters J.F., Vergili T., Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan curve theorem, Applied General Topology, 24(1), 25-45, 2023.
  24. Preuss G., Theory of Topological Structures, D. Reidel Publishing Company, 1988.
  25. Scoville N.A., On the Lusternik-Schnirelmann category of a simplicial map, Topology and Its Applications, 216, 116-128, 2017.
  26. Sharma P.L., Proximity bases and subbases, Pacific Journal of Mathematics, 37(2), 515-526, 1971.
  27. Smirnov J.M., On proximity spaces, Matematicheskii Sbornik, 31(73)(3), 543-574, 1952.
  28. Switzer R.M., Algebraic Topology: Homology and Homotopy, Classics in Mathematics, 212, Springer, 2002.
  29. Vergili T., Connectedness and Lusternik-Schnirelmann categories of the spaces of persistence modules, New York Journal of Mathematics, 26, 598-606, 2020.