Structural properties of pairwise difference Lindelöf spaces: statistical applications in data analysis
Résumé
This paper formalizes D-Lindelöf spaces in bitopological spaces, extending Lindelöf concept through D-sets. Key contributions include proving countable spaces are generally pairwise D-Lindelöf; establishing that pairwise D-Lindelöfness implies 1- Lindelöfness; demonstrating closure of D-sets under finite intersections; and analyzing
preservation under pairwise continuous surjections, D-irresolute functions, and perfect functions. The inheritance conditions for 1-closed subspaces are determined, with limitations clarified by counterexamples. Beyond theory, we demonstrate statistical applications: (i) D-set classification for mixed-feature data, (ii) convergence analysis under dual topologies, (iii) D-compactness in probability measures, and (iv) parameter identification modeling.
Téléchargements
Références
Alrababah, R., Amourah, A., Salah, J., Ahmad, R., & Atoom, A. A. (2024). New Results on Difference Paracompactness in Topological Spaces. European Journal of Pure and Applied Mathematics, 17(4), 2990-3003.
Arhangel’skii, A. V. (2005). D-spaces and covering properties. Topology and its Applications, 146, 437-449.
ATOOM, A. A., & BANI-AHMAD, F. (2024). Between Pairwise--Perfect Functions and Pairwise-t--Perfect Functions. Journal of applied mathematics & informatics, 42(1), 15-29.
Atoom, A. A., Bani Abdelrahman, M. A.(2025). Difference Compactness in Bitopological Spaces: Foundations from Difference Sets and Dual Views with Applications. Boletim da Sociedade Paranaense de Matemática, 43.
Atoom, A. A., Bani Abdelrahman, M. A.(2026). Innovative aggregation methods: semi-Lindelöf perfect functions and semi-perfect functions in bitopological spaces, together with their utilization. International Journal of Maps in Mathematics, 9(1).
Atoom , A., Qoqazeh, H., Abdelrahman, M. A. B., Hussein, E., Mahmoud, D. A., & Owledat, A. A. (2025). A Spectrum of Semi-Perfect Functions in Topology: Classification and Implications. WSEAS Transactions on Mathematics, 24, 347-357.
Atoom, A. A., Al-Otaibi, M., Mo’men, G. G., Qoqazeh, H., & AlKhawaldeh, A. F. O. (2025). Exploring the Role of [d,e]-Lindelöf Spaces: Theoretical Insights and Practical Implications. International Journal of Analysis and Applications, 23, 136-136.
Atoom, A., Qoqazeh, H., & Abu Alkishik, N. (2024). Study the structure of difference Lindel¨of topological spaces and their properties. Journal of Applied Mathematics & Informatics, 42(3), 471–481.
Atoom, A. A., et al. (2025). Exploring the difference ParaLindelöf in topological spaces. International Journal of Analysis and Applications, 23, 24.
Atoom, A. A., et al. (2024). A new outlook on omega closed functions in bitopological spaces and related aspects. European Journal of Pure and Applied Mathematics, 17(4), 2574–2585.
Bani-Ahmad, F., Alsayyed, O., & Atoom, A. A. (2022). Some new results of difference perfect functions in topological spaces. AIMS Mathematics, 7(11), 20058–20065.
Borges, C. R., & Wehrly, A. C. (1991). A study of D-spaces. In Topology Proc (Vol. 16, pp. 7-15).
Eisworth, T. (2007). On D-spaces. In Open problems in topology II (pp. 129-134). Elsevier.
Fletcher, P., Hoyle, H. B., III, & Patty, C. W. (1969). The comparison of topologies. Duke Mathematical Journal, 36, 325–331.
Mahmood, S. I. and Abdul-Hady, A. A. (2019) Weak soft (1 , 2) ∗ − ˜Dω-sets and weak soft (1 , 2) ∗ − ˜Dω-separation axioms in soft bitopological spaces, Al-Nahrain journal of science,22(2), pp 49-58.
Padma, P., Dhanabalan, A. P., & Kumar, S. U. (2017). Q* D-sets in topological space. Acta ciencia indica, 2, 135-140.
Peng, L. X. (2017). Some sufficiency conditions for D-spaces, dually discrete and its applications. Topology and its Applications, 217, 81-94.
Thomas, J., & John, S. J. (2015). Properties of D-compact spaces in generalized topological spaces. Annals of Pure and Applied Mathematics, 9(1), 73-80.
Tong, J. (1982). A separation axioms between T0 and T1. Annales de la Société Scientifique de Bruxelles, 96(2), 85–90.
Zhang, H., & Shi, W. X. (2012). A note on D-spaces. Topology and its Applications, 159(1), 248-252.
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
Ce travail est disponible sous la licence Creative Commons Attribution 4.0 International .
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
