Difference Compactness in Bitopological Spaces: Foundations from Difference Sets and Dual Views with Applications
D-Compactness in Bitopological Spaces
Resumo
The notion of $D-$compactness in bitopological spaces, an inventive generalization that combines the structural variety of dual topologies with the adaptability of $D$-sets, is introduced and examined in this paper. Through a series of theorems and examples, we characterize pairwise $D-$compact spaces and establish basic properties of pairwise $D$-sets. A detailed analysis is conducted of the relationship between pairwise $D-$compactness and classical concepts like pairwise compactness and pairwise Hausdorff spaces. We investigate how pairwise $D-$compactness behaves under different topological operations. In addition to expanding the theoretical framework of bitopological spaces, this work lays the groundwork for applications in computational topology, Rough Set Theory, and multi-criteria decision making.
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Referências
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\bibitem{atoom2024} Atoom, A. A., Qoqazeh, H., \& Abu-alkishik, N. (2024). Study the Structure of Difference Lindelöf Topological Spaces and Their Properties. Journal of applied mathematics \& informatics, 42(3), 471-481.
\bibitem{atoom2025} Atoom, A. A., Alrababah, R., Alholi, M., Qoqazeh, H., Alnana, A., \& Mahmoud, D. A. (2025). Exploring the Difference Paralindelöf in Topological Spaces. International Journal of Analysis and Applications, 23, 24-24.
\bibitem{alrababah} 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.
\bibitem{Almuhur} Almuhur, E., Qoqazeh, H., Atoom, A., \& Abu-alkishik, Nabeela. PAIRWISE 𝐃𝛂− OPEN AND 𝐃𝛂− CLOSED FUNCTIONS IN BITOPOLOGICAL SPACES.
\bibitem{atoom2018} Atoom, A. (2018). Study of pairwise− ω− compact spaces. Global Journal of Pure and Applied Mathematics, 14(11), 1453-1459.
\bibitem{abualkishik2024} Atoom, A. A., Qoqazeh, H., \& Abu-Alkishik, N. (2024). Study the Structure of Difference Lindelöf Topological Spaces and Their Properties. Journal of applied mathematics \& informatics, 42(3), 471-481.
\bibitem{Birsan1969} Birsan, T. (1969). Compacité dans les espaces bitopologiques. An. st. Univ. Iasi, s. I. a., Matematica, 15, 317-328.
\bibitem{Fletcher1969} Fletcher, P., Hoyle, III, H. B., \& Patty, C. W. (1969). The comparison of topologies.
\bibitem{fora&hdeib1983} Fora, A. A., \& Hdeib, H. Z. (1983). On pairwise Lindelöf spaces. Revista colombiana de matematicas.
\bibitem{dorsett1981} Dorsett, C. (1981). Semi convergence and semi compactness. Indian J. Mech. Math, 19(1), 11-17.
\bibitem{engelking1989} Engelking, R. (1989). General topology. Sigma series in pure mathematics, 6.
\bibitem{hanna1984} Hanna, F. A. D. (1984). CHARACTERIZATIONS OF SEMI COMPACT SPACES.
\bibitem{kelly1963} Kelly, J. (1963). Bitopological spaces. Proceedings of the London Mathematical Society, 3(1), 71-89.
\bibitem{Kim1968} Kim, Y. W. (1968). Pairwise compactness. Publ. Math. Debrecen, 15, 87-90.
\bibitem{lavine1963} Levine, N. (1963). Semi-Open Sets and Semi-Continuity in Topological Spaces. American Mathematical Monthly, 70, 36-41.
\bibitem{mahmoud2019} Mahmood, S. I. and Abdul-Hady, A. A. (2019) Weak soft $\left(
1~,~2\right) ^{\ast }-\tilde{D}_{\omega }$-sets and weak soft $\left(
1~,~2\right) ^{\ast }-\tilde{D}_{\omega }$-separation axioms in soft
bitopological spaces, Al-Nahrain journal of science,22(2), pp 49-58.
\bibitem{mohsin2019} Mohsin, R. (2019). Pairwise locally compact and compactification in. Journal of Kufa for Mathematics and Computer, 6(2).
\bibitem{mukharjee2013} Mukharjee, A., \& Bose, M. K. (2013). On nearly pairwise compact spaces. Kyungpook mathematical journal, 53(1), 125-133.
\bibitem{pahk1971} Pahk, D. H., \& Choi, B. D. (1971). Notes on pairwise compactness. Kyungpook Mathematical Journal, 11(1), 45-52.
\bibitem{reilly1972} Reilly, I. L. (1972, January). Bitopological local compactness. In Indagationes Mathematicae (proceedings) (Vol. 75, No. 5, pp. 407-411). North-Holland.
\bibitem{Tong1982} Tong, J. (1982), A separation axioms between $T_{0}$ and $T_{1}$, Ann.
Soc. Sci. Bruxelles.96, $\left( II\right) $ , 85-90.
\bibitem{vithya2018} Vithya, N. (2018), On $b^{\sharp }D$ sets and associated separation
axioms , Taga journal , 14 , pp 3314-3326.
Publicado
2025-09-01
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