cc4173cd-fcf9-4ef8-ae12-6d75d6ef1b2b20210402030112148naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=6985292021292021810.46300/91019.2021.8https://www.naun.org/cms.action?id=23293Raja MohammadLatifDepartment of Mathematics and Natural Sciences Prince Mohammad Bin Fahd University P.O. Box 1664 Al – Khobar 31952 Saudi ArabiaIn 1968 Velicko [30] introduced the concepts of δ-closure and δ-interior operations. We introduce and study properties of δ-derived, δ-border, δ-frontier and δ-exterior of a set using the concept of δ-open sets. We also introduce some new classes of topological spaces in terms of the concept of δ-D- sets and investigate some of their fundamental properties. Moreover, we investigate and study some further properties of the well-known notions of δ-closure and δ-interior of a set in a topological space. We also introduce δ-R0 space and study its characteristics. We also introduce δ-R0 space and study its characteristics. We introduce δ-irresolute, δ-closed, pre-δ-open and pre -δ-closed mappings and investigate properties and characterizations of these new types of mappings and also explore further properties of the well-known notions of δ-continuous and δ-open mappings.292021292021122https://www.naun.org/main/NAUN/puremath/2021/a022019-001(2021).pdf10.46300/91019.2021.8.1https://www.naun.org/main/NAUN/puremath/2021/a022019-001(2021).pdfS.P. Arya and M. Deb, On θ-continuous mappings, Math. Student 42(1974), 81-89. C.W. Baker, On θ-C open sets, Int. J. Math. Math. Sci. 15(1992), no. 2, 255-259. M. Caldas, D.N. Georgian, S. Jafari and T. Noiri, More on δ-semiopen sets, Note di Mathematica, 22(2) (2003/2004), 113-126. 10.4995/agt.2005.1964J. Cao, M. Ganster, I. Reilly and M. Steiner, δ-closure,θ-closure, and generalized closed sets, Applied General Topology, 6(1) (2005), 79-86. 10.2140/pjm.1975.59.407R.F. Dickman, Jr. and J.R. Porter, θ-closed subsets of Hausdorff spaces, Pacific J. Math. 59(1975), 407-415. 10.1215/ijm/1256049499R.F. Dickman, Jr. and J.R. Porter, θ-perfect and θ-absolutely closed functions, Illinois J. Math. 21(1977), 42-60. J. Dontchev, H. Maki, Groups of θ- generalized homeomorphisms and the digital line, Topology and its Applications, 20(1998), 1- 16. J. Dontchev and H. Maki, On θ-generalized closed sets, Int. J. Math. Math. Sci. 22(1999), no. 2, 239-249. E. Ekici, (δ-pre,s)-continuous functions, Bulletin of the Malaysian Mathematical Sciences Society, Second Series 27(2004), no. 2, 237-251. E. Ekici, On δ-semiopen sets and a generalization of functions, Bol. Soc. Mat. (38) vol. 23 (1-2) (2005), 73-84. Ganster1988 : M. Ganster, T. Noiri, I.L. Reilly, Weak and strong forms of θ-irresolute functions, J. Inst. Math. Comput. Sci. 1(1) (1988), 19-29. S. Jafari, Some properties of θ-continuous functions, Far East J. Math. Sci. 6(5) (1998), 689-696. D.S. Jankovic, On some separation axioms and θ-closure, Mat. Vesnik 32(4) (1980), 439- 449. D.S. Jankovic, θ-regular spaces, Internat. J. Math. & Math. Sci. 8(1986), 515-619. J.E. Joseph, θ-closure and θ-subclosed graphs, Math., Chronicle 8(1979), 99-117. A. Kilicman, Z. Salleh, Some results on (δ- pre,s)-continuous functions, International Journal Math. Mat. Sci. 2006(2006), 1-11. M.M. Kovar, On θ-regular spaces, Internat. J. Math. & Math. Sci. 17(1994), 687-692. R.M. Latif, On characterizations of mappings, Soochow Journal of Mathematics, 19(4) (1993), 475-495. R.M. Latif, Semi-convergence of filters and nets, Mathematical Journal of Okayama University, 41(1999), 103-109. R.M. Latif, Characterizations and applications of γ-open sets, Soochow Journal of Mathematics, 32(3) (2006), 369-378. N. Levine, Semi-open sets and semicontinuity in topological space, Amer. Math. Monthly 70 (1963), 36-41. P.E. Long, L.L. Herrington, The   - topology and faintly continuous functions, Kyungpook Math. J. 22(1982), 7-14. T. Noiri, On δ-continuous functions, J. Korean Math. Soc., 16 (1980), 161-166. 10.1016/s0166-8641(01)00180-8T. Noiri, S. Jafari, Properties of (θ,s)- continuous functions, Topology and its Applications, 123(1) (2002), 167-179. J.H. Park, B. Y. Lee and M.J. Son, On δ- semiopen sets in topological space, J. Indian Acad. Math., 19(1), (1997), 59-67. S. Raychaudhuri and M.N. Mukherjee, On δ-almost continuity and δ-preopen sets, Bulletin of the Institute of Mathematics, Academia Sinica 21 (1993), no. 4, 357-366. M. Saleh, Some applications of δ-sets to Hclosed spaces, Q&A Topology 17(1999), 203- 211. M. Saleh, On super and δ-continuities, Mathematics and Mathematics Education, World Scientific, 2002, 281-291. M. Saleh, On θ-closed sets and some forms of continuity, Archivum Mathematicum (BRNO) 40(2004), 383-393. N.V. Velicko, H-closed topological spaces, Mat. Sb., 98-112; English transl. (2), in Amer. Math. Soc. Transl., 78(1968), 102-118. S. Willard, General Topology, Addison- Wesley Publishing Company, Inc., 1970.