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Journals / TWMS Journal of Applied and Engineering Mathematics / 2026 / Volume: 16 Issue: 10

TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES

Raghad Almohammed(General Directorate of Education of Karbala, Ministry of Education)0000-0003-0559-0436
Dunia M.k. Al-Ftlawy(General Directorate of Education of Karbala, Ministry of Education)0009-0007-3600-1389
Nadiha Abed Habeeb(College of Education Al-Zahraa, University for Women)0009-0008-3552-744X
Luay Al. Swidi(College of Education for Pure Sciences, University of Babylon)0000-0001-5641-1616
Pages
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Publication Date
2026-10-06
DOI
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AI Summary & Key Takeaways

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

We close with a discussion of the broader implications of these results for the relationship between cluster theory and generalized compactness.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this work we introduce a family of subsets of a proximity space, called the typical family, defined as the complement of a cluster family, and we study its main properties and its relation to the cluster concept.
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Key Findings & Data:We illustrate the framework with a fully worked finite numeric example.
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Conclusion & Impact:We close with a discussion of the broader implications of these results for the relationship between cluster theory and generalized compactness.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Cluster Family · Ideal · Proximity

Abstract

In this work we introduce a family of subsets of a proximity space, called the typical family, defined as the complement of a cluster family, and we study its main properties and its relation to the cluster concept. We show explicitly that the typical family is, up to the language of proximity spaces, the ideal that is dual to the associated cluster in the classical sense of Choquet's grill ideal duality, and we compare it in detail with ideals, filters, grills, and bornologies, establishing a necessary and sufficient condition under which it is a bornology. Building on this family we define a generalized compactness notion, called $T$-compactness, relate it to the classical notion of compactness with respect to an ideal, and establish preservation results for finite unions, closed subsets, and via a pushforward construction introduced here images under proximity isomorphisms. We illustrate the framework with a fully worked finite numeric example. We close with a discussion of the broader implications of these results for the relationship between cluster theory and generalized compactness.

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Cite This Article

Almohammed, R., Al-Ftlawy, D. M., Habeeb, N. A., Swidi, L. A. (2026). TYPICAL FAMILIES AND $T$-COMPACTNESS IN PROXIMITY SPACES. TWMS Journal of Applied and Engineering Mathematics, Volume: 16 Issue: 10. https://ajindex.com/articles/1604191

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References (32)

  1. Efremovic, V. A., (1951), Infinitesimal spaces, Doklady Akademii Nauk SSSR, 76(3), pp. 341-343.
  2. Lodato, M. W., (1964), On topologically induced generalized proximity relations, Proceedings of the American Mathematical Society, 15(3), pp. 417-422.
  3. Leader, S., (1967), Local proximity spaces, Mathematische Annalen, 169, pp. 275-281.
  4. Naimpally, S. and Warrack, B., (1970), Proximity Spaces, Cambridge University Press.
  5. Peters, J. F., (2014), Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Springer.
  6. Peters, J. F. and Guadagni, C., (2014), Strongly near proximity and hyperspace topology, General Mathematics Notes, 21(2), pp. 70-81.
  7. Di Concilio, A., (2009), Proximity: A powerful tool in extension theory, function spaces, hyperspaces, Boolean algebra, and point-free geometry, Contemporary Mathematics, 486, pp. 31-56.
  8. Beer, G., (1993), Topologies on Closed and Closed Convex Sets, Springer.
  9. Csaszar, A., (2002), Generalized topology, generalized continuity, Acta Mathematica Hungarica, 96(4), pp. 351-357.
  10. Hamlett, T. R. and Jankovic, D., (1990), Compactness with respect to an ideal, Bollettino U.M.I., 7(4-B), pp. 849-861.
  11. Newcomb, R. L., (1986), Topologies generated by ideals, Pacific Journal of Mathematics, 123(1), pp. 215-220.
  12. Abed Noor, E. J. and Mahdi Altalkany, Y. K., (2024), Characterization of new types of continuous function via ideal topological space, ICPAS, 232.
  13. Nasef, A. A. and Noiri, T., (2008), Some forms of compactness in ideal topological spaces, Acta Mathematica Hungarica, 118(3), pp. 223-234.
  14. Jankovic, D. and Hamlett, T. R., (1990), New topologies from old via ideals, American Mathematical Monthly, 97(4), pp. 295-310.
  15. Dontchev, J., Ganster, M. and Rose, D., (1999), Ideal resolvability, Topology and its Applications, 93(1), pp. 1-16.
  16. Abd El-Monsef, M. E., Lashien, E. F. and Nasef, A. A., (1992), On I-open sets and I-continuous functions, Kyungpook Mathematical Journal, 32(1), pp. 21-30.
  17. Kareem, M. K. and Jabar, L. A. A., (2026), f-Focal and f-nested sets, American Institute of Physics Conference Series, 3393(1).
  18. Peters, J. F., (2018), Computational proximity: Excursions in the topology of digital images, Intelligent Systems Reference Library, Springer.
  19. Guadagni, C. and Peters, J. F., (2016), Descriptive proximities I: Properties and interplay between classical proximities and overlap, Mathematics in Computer Science, 10(1), pp. 91-106.
  20. Ivanova, M., (2020), Descriptive proximity spaces and applications, Filomat, 34(12), pp. 3921-3935.
  21. Alimohammady, M. and Roohi, M., (2009), Fixed point in fuzzy and proximity spaces, Chaos, Solitons and Fractals, 42(1), pp. 468-472.
  22. Gunduz, C., Bayramov, S., Zeynalov, J.,(2024), Separation Axioms In Bipolar Soft Topological Spaces, TWMS J. Pure Appl. Math., 15(1), pp. 15-25.
  23. Peters, J. F. and Wasilewski, P., (2009), Foundations of near sets, Information Sciences, 179(18), pp. 3091-3109.
  24. Al-Hawary, T. and Noiri, T., (2020), Generalized compactness and ideal topological spaces, European Journal of Pure and Applied Mathematics, 13(3), pp. 590-603.
  25. Roy, B. and Mukherjee, M., (2021), On a class of generalized compact spaces, Applied General Topology, 22(1), pp. 53-68.
  26. Kocinac, L. D. R., (2011), Selection principles and generalized compactness, Topology and its Applications, 158(12), pp. 1443-1452.
  27. Almohammed, R. and Jabar, L. A. A., (2024), Dismountable and non-Dismountable Spaces via Proximity Space, Baghdad Science Journal.
  28. AL-Swidi, L. A. A. and Mutakek, A. Y. K., (2025), JD-paracompact space, Journal of Interdisciplinary Mathematics, 28(1).
  29. Peters, J. F., (2022), Proximity spaces and applications in computational topology, Axioms, 11(5).
  30. Di Concilio, A. and Guadagni, C., (2022), Proximity structures and generalized topological systems, Mathematics, 10(14).
  31. Roy, B. and Sen, R., (2023), Ideal compactness and generalized topological structures, Demonstratio Mathematica, 56(1).
  32. Choquet, G., (1947), Sur les notions de filtre et de grille, Comptes Rendus de l’Acad´emie des Sciences Paris, 224, pp. 171-173.