International Journal of Nonlinear Analysis and Applications

Pairwise compactness in bi-isotonic spaces

Articles in Press

Document Type : Research Paper

Authors

Sakarya University, Faculty of Science, Department of Mathematics, 54050 Sakarya, Turkey

10.22075/ijnaa.2024.32706.4867

Abstract

In this article, we have introduced the notion of pairwise compactness in bi-isotonic spaces via finite intersection property and pairwise open cover. Moreover, we have given pairwise compactness of bi-isotonic subspaces with both reduced closure and interior functions. We have characterized pairwise compactness by the neighborhood concept; however, the axioms of bi-isotonic spaces are insufficient to prove the theorem, and we have studied this in bi-closure spaces. For similar reasons, occasionally, bi-closure spaces have been considered with additional explanations, even if some concepts related to compactness have been naturally extended to bi-isotonic spaces. Additionally, interesting results have been obtained considering the pairwise Hausdorffness and compactness relationship. It has also been observed that resembling cross-relationships exist for closed subsets of pairwise compact spaces. Finally, it has been observed that the pairwise compactness of bi-closure spaces is preserved under bi-continuity.

Keywords

[1] T. Bırsan, Compacite dans les espaces bitopologiques, An. Sti. Univ. Al. I. Cuza Iasi Sect. I Mat. (N.S.) 15 (1969), 317–328.
[2] C. Boonpok, ∂−closed sets in biclosure spaces, Acta Math. Univ. Ostrav. 17 (2009), 51–66.
[3] C. Boonpok, On closed maps in bi Cech closure spaces, Int. Math. Forum 4 (2010), 2161–2167.
[4] S. Ersoy and A.A. Acet, Pairwise connectedness in bi-isotonic spaces, Palest. J. Math. 11 (2022), no. 2, 342––351.
[5] S. Ersoy and N. Erol, Separation axioms in bi-isotonic spaces, Sci. Math. Jpn. 83 (2020), no. 3, 225–240.
[6] P. Fletcher, H.B. Hoyle, and C.W. Patty, The comparison of topologies, Duke Math. J. 36 (1969), 325–331.
[7] E.D. Habil and K.A. Elzenati, Connectedness in isotonic spaces, Turk. J. Math. 30 (2006), no. 3, 247—262.
[8] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13 (1963), no. 1, 71–89.
[9] Y.W. Kim, Pairwise compactness, Publ. Math. Debrecen 15 (1968), no. 1-4, 87–90.
[10] K. Kuratowski, Topology, vol. I, Academic Press; Warszawa: PWN—Polish Scientific Publishers, New York-London, 1966.
[11] L. Motchane, Sur la notion d’espace bitopologique et sur les espaces de Baire, C. R. Acad. Sci. Paris 224 (1957), 3121-3124.
[12] A. Mukharjee and M.K. Bose, Some results on nearly pairwise compact spaces, Bull. Malays. Math. Sci. Soc. 39 (2016), 933—940.
[13] D.H. Pahk and B.D. Choi, Notes on pairwise compactness, Kyungpook Math. J. 11 (1971), 45–52.
[14] J. Saegrove, Pairwise complete regularity and compactification in bitopological spaces, J. London Math. Soc. (2) 7 (1973), 286––290.
[15] B.M.R. Stadler and P.F. Stadler, Basic properties of closure spaces, J. Chem. Inf. Comput. Sci. 42 (2002), 577-585.
[16] B.M.R. Stadler and P.F. Stadler, Higher separation axioms in generalized closure spaces, Comment. Math. (Prace Mat.) 43 (2003), 257–273.
[17] J. Swart, Total disconnectedness in bitopological spaces and product bitopological spaces, Indag. Math. 33 (1971), 135–145.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 27 June 2024
  • Receive Date: 19 December 2023
  • Revise Date: 17 January 2024
  • Accept Date: 05 April 2024